https://comb-opt.azaruniv.ac.ir/ Communications in Combinatorics and Optimization en daily 1 Tue, 01 Sep 2026 00:00:00 +0330 Tue, 01 Sep 2026 00:00:00 +0330 https://comb-opt.azaruniv.ac.ir/article_14870.html In this paper we introduce a new domination problem strongly related to the following one recently proposed by Broe, Chartrand and Zhang. One says that a vertex $v$ of a graph $\Gamma$ labeled with an integer $\ell$ dominates the vertices of $\Gamma$ having distance $\ell$ from $v$. An irregular dominating set of a given graph $\Gamma$ is a set $S$ of vertices of $\Gamma$, having distinct positive labels, whose elements dominate every vertex of $\Gamma$. Since it has been proven that no connected vertex transitive graph admits an irregular dominating set, here we introduce the concept of an \emph{extended} irregular dominating set, where we admit that precisely one vertex, labeled with 0, dominates itself. Then we present existence or non existence results of an extended irregular dominating set $S$ for several classes of graphs, focusing in particular on the case in which $S$ is as small as possible. We also propose two conjectures.    https://comb-opt.azaruniv.ac.ir/article_14871.html A weak signed double Roman dominating function (WSDRDF) of a graph $G$ with vertex set $V(G)$ is defined as a function $f:V(G)\rightarrow\{-1,1,2,3\}$ having the property that $\sum_{x\in N[v]}f(x)\ge 1$ for each $v\in V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSDRDF is the sum of its function values over all vertices. The weak signed double Roman domination number of $G$, denoted by $\gamma_{wsdR}(G)$, is the minimum weight of a WSDRDF in $G$. We initiate the study of the weak signed double Roman domination number, and we present different sharp bounds on $\gamma_{wsdR}(G)$. In addition, we determine the weak signed double Roman domination number of some classes of graphs. https://comb-opt.azaruniv.ac.ir/article_14890.html The locating rainbow connection number of a graph is defined as the minimum number of colors required to color vertices such that for every two vertices there exists a rainbow vertex path and every vertex has a distinct rainbow code. This rainbow code signifies a distance between vertices within a given set of colors in a graph. This paper aims to determine the locating rainbow connection number for vertex-transitive graphs. Three main theorems are derived, focusing on the locating rainbow connection number for cycles, $(n-2)$-regular graphs, and complement of cycles $\overline{C_n}$. https://comb-opt.azaruniv.ac.ir/article_14842.html Let $G_S$ be a graph obtained by attaching a self-loop to each vertex of $S\subseteq V$  of a graph $G(V,E)$. The Seidel matrix of $G_S$ is $S(G_S)=[s_{ij}]$, where $s_{ij}=-1$ if $v_i$ and $v_j$ are adjacent and $v_i\in S$, $s_{ij}=1$ if $v_i$ and $v_j$ are non-adjacent, and it is zero if $i=j$ and $v_i\not\in S$.     If $\theta_i(G_S)\,,\,i=1,2,\ldots,n$, are the eigenvalues of the Seidel matrix, then the Seidel energy of the graph $G_S$, containing $n$ vertices and $\sigma$ self-loops, is defined as $\sum_{i=1}^n \left|\theta_i(G_S)+\frac{\sigma}{n}\right|$. In this paper, some basic properties of Seidel energy of graphs containing self-loops are established. https://comb-opt.azaruniv.ac.ir/article_14891.html Recently a new vertex-degree based molecular structure descriptor was defined as Sombor index. For a simple graph $G$, the Sombor index of $G$, denoted by $SO(G)$, is defined as $\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2},$ where $d_v$ is the degree of $v$. In this paper we study the Sombor index of many kinds of product of graphs, such as join of graphs, Cartesian product of graphs, tensor product of graphs, and lexicographic product of graphs. We obtain some formulas for the Sombor index of these product of graphs. https://comb-opt.azaruniv.ac.ir/article_14847.html The $Q$-eigenvalues are the eigenvalues of the signless Laplacian matrix $Q(G)$ of a graph $G$, and the largest $Q$-eigenvalue is known as the $Q$-spectral radius $q(G)$ of $G$. The edge-degree of an edge is defined as the number of edges adjacent to it. In this article, we characterize the structure of simple connected graphs having integral $Q$-spectral radius. We show that the necessary and sufficient condition for such graphs to contain either a double star $\mathcal{S}_{r}^{2}$ or its variation $\mathcal{S}_{r}^{2,1}$ (having exactly one common neighbor between the central vertices) as a subgraph is that the maximum edge-degree is $2r$, where $r= q(G) -3$. In particular, we characterize all graphs that contain only double star as a subgraph when $q(G)$ equals $8$ and $9$. Further, we characterize all the connected edge-non-regular graphs with a maximum edge-degree equal to $4$ whose minimum  $Q$-eigenvalue does not belong to the open interval $(0,1)$ and has an integral $Q$-spectral radius. https://comb-opt.azaruniv.ac.ir/article_14896.html Let $G$ be a graph. An {\it $r\!$-dynamic $k\!$-coloring} of $G$ is a proper $k\!$-coloring of $G$ such that every vertex $v$ in $V(G)$ has neighbors in at least $\min\{r,d_G(v)\}$ different color classes. The {\it $r\!$-dynamic chromatic number} of $G,$ denoted by $\chi_r(G),$ is the least $k$ such that $G$ has an $r\!$-dynamic $k\!$-coloring. We determine the $r\!$-dynamic chromatic number of the corona product $G\odot H$ of graphs $G$ and $H,$ in terms of the dynamic chromatic numbers of $G$ and $H.$ https://comb-opt.azaruniv.ac.ir/article_14878.html The Sombor index is a vertex degree-based topological index which it was de ned by Ivan Gutman in 2021. We study the Sombor index and the multiplicative Sombor index on  some products of graphs, crown graphs, shelf graphs, Ice-cream graphs, helm graphs, ower graphs, generalized Sierpiński graphs, $t$-Mycielskian graphs, $t$-ciclo graphs, and $t$-estella graphs. Then we provide some upper and lower bounds for them. https://comb-opt.azaruniv.ac.ir/article_14895.html Let $H$ be a digraph possibly with loops, and $D$ a multidigraph without loops. An $H$-coloring of $D$ is a function $c: A(D) \rightarrow V(H)$. We say that $D$ is an $H$-colored multidigraph whenever we are taking a fixed $H$-coloring of $D$. A trail $W=(v_0,e_0,v_1,e_1,v_2,\ldots,v_{n-1},e_{n-1},$ $v_n)$ in $D$ is an $H$-trail if and only if $(c(e_i),c(e_{i+1}))$ is an arc in $H$, for each $i \in \{0,\ldots,n-2\}$. We say that an $H$-colored multidigraph is $H$-trail-connected if and only if there is an $H$-trail starting with arc $f_1$ and ending with arc $f_2$, for any pair of arcs $f_1$ and $f_2$ in $D$. Let $D$ be an $H$-colored multidigraph and $u$ a vertex of $D$, the auxiliary digraph $D_u$ is the digraph of allowed transition throughout $u$.In this paper we give the following characterization: Let $D$ be an $H$-colored multidigraph such that the underlying graph of $D_u$ is a disjoint union of complete bipartite graphs, for every $u \in V(D)$. Then $D$ has a Euler $H$-trail if and only if $D$ is $H$-trail-connected and, for every $u \in V(D)$, the underlying graph of $D_u$ has a perfect matching. As a consequence we obtain the well-known characterization of the 2-arc-colored multidigraphs containing properly colored Euler trail. Finally, we give an infinite family of digraphs $H$ such that for every multidigraph $D$ without isolated vertices, and every $H$-coloring of $D$, the underlying graph of $D_u$ is a disjoint union of complete bipartite graphs and, possibly, isolated vertices, for every $u \in V(D)$. https://comb-opt.azaruniv.ac.ir/article_14848.html Consider a graph $\sigma$(V, E) with nodes V and edges E is a connected graph with some pebbles scattered over its nodes V. By removal of two pebbles from one node and placing one pebble to an adjacent node is a pebbling move. A monophonic pebbling number, $\lambda_{M}(\sigma, v)$, of a node v of a graph $\sigma$ is the least number $m$ such that minimum of one pebble could be shifted to v by a sequence of pebbling shifts for any distribution of $\lambda_{M}(\sigma, v)$ pebbles on the nodes of $\sigma$ using monophonic path. A link between the nodes x and y is an x-y path which consists of no chords and is monophonic. The monophonic pebbling number of a graph $\sigma$ is the highest $\lambda_{M}(\sigma, v)$ among all the nodes notated as $\lambda_{M}(\sigma)$. For the first time, we calculate the monophonic pebbling number on families of neural networks such as probabilistic neural networks(PNNs),  convolutional neural networks(CVNNs), modular neural networks(MNNs), generalized regression neural networks(GRNNs) and Hopfield neural networks(HNNs) and discuss their applications. We give the generalized algorithm to find the monophonic pebbling number of any graph $\sigma$. https://comb-opt.azaruniv.ac.ir/article_14897.html Consider a simple connected graph G with the vertex set V (G) and edge set E(G). The Mostar index M◦(G) of G is defined as M◦(G) = 􏰌e=xy∈E(G) |nx − ny|, where nx and ny represent the number of vertices that lie closer to x than to y and the number of vertices that lie closer to y than to x, respectively. In this paper, we prove that if G is a tree, then M◦(LG) < M◦(G), where LG is the line graph. In order to provide an example supporting this result, we develop three algorithms (and implement them using Python) to calculate the Mostar index of trees of order at most 8 and their line graphs. https://comb-opt.azaruniv.ac.ir/article_14872.html Let $G$ be a simple graph. The dominated coloring of a graph $G$ is a proper coloring of $G$ such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of $G$ is called the dominated chromatic number of $G$, denoted by $\chi_{dom}(G)$. The current study is devoted to investigate the dominated chromatic number of Helm graphs and some  kinds of its the generalizations. https://comb-opt.azaruniv.ac.ir/article_14898.html For a lattice $L$, the strongly annihilator ideal graph of $L$ is denoted by $SAnnIG(L)$. It is a graph with the vertex set, which consists of all ideals in $L$ that have nontrivial annihilators such that any two distinct vertices $I$ and $J$ are adjacent in $SAnnIG(L)$ if and only if the annihilator of $I$ contains a nonzero element of $J$ and the annihilator of $J$ contains a nonzero element of $I$. In this paper, we determine the radius, circumference, and domination number of $SAnnIG(L)$. We obtain necessary and sufficient conditions for $SAnnIG(L)$ to be in the class of paths, cycles, unicyclic, triangle-free, trees, complete multipartite, split or claw-free graphs. Among other results, we study the affinity between the strongly annihilator ideal and the annihilator ideal graph of a lattice. https://comb-opt.azaruniv.ac.ir/article_14850.html Let $\Gamma$ be a finite group with $T_\Gamma=\{t\in \Gamma \mid t\ne t^{-1} \}$. The inverse graph of $\Gamma$, denoted by $IG(\Gamma)$, is a graph whose vertex set is $\Gamma$ and two distinct vertices, $u$ and $v$, are adjacent if $u*v\in T_\Gamma$ or $v*u\in T_\Gamma$. In this paper, we study the rainbow connection number of the connected inverse graph of a finite group $\Gamma$, denoted by $rc(IG(\Gamma))$, and its relationship to the structure of $\Gamma$. We improve the upper bound for $rc(IG(\Gamma))$, where $\Gamma$ is a group of even order. We also show that for a finite group $\Gamma$ with a connected $IG(\Gamma)$, all self-invertible elements of $\Gamma$ is a product of $r$ non-self-invertible elements of $\Gamma$ for some $r\leq rc(IG(\Gamma))$. In particular, for a finite group $\Gamma$, if $rc(IG(\Gamma))=2$, then all self-invertible elements of $\Gamma$ is a product of two non-self-invertible elements of $\Gamma$. The rainbow connection numbers of some inverse graphs of direct products of finite groups are also observed. https://comb-opt.azaruniv.ac.ir/article_14914.html Let $G=(V,E)$ be a connected graph and $d_{G}(u,v)$ be the shortest distance between the vertices $u$ and $v$ in $G$. A set $S=\{s_{1},s_{2},\dots,s_{n}\}\subset V(G)$ is said to be a {\em resolving set} if for all distinct vertices $u,v$ of $G$, there exists an element $s\in S$ such that $d_{G}(s,u)\neq d_{G}(s,v)$. The minimum cardinality of a resolving set for a graph $G$ is called the metric dimension of $G$, and it is denoted by $\beta{(G)}$. A resolving set having $\beta{(G)}$ number of vertices is named as metric basis of $G$. The metric dimension problem is to find a metric basis in a graph $G$, and it has several real-life applications in network theory, telecommunication, image processing, pattern recognition, and many other fields. In this article, we consider cube of trees $T^{3}=(V, E)$, where any two vertices $u,v$ are adjacent if and only if the distance between them is less than or equal to three in $T$. We establish the necessary and sufficient conditions for a vertex subset of $V$ to become a resolving set for $T^{3}$. This helps to determine the tight bounds (upper and lower) on the metric dimension of $T^{3}$. Then, for certain well-known cube of trees, such as caterpillars, lobsters, spiders, and $d$-regular trees, we establish the boundaries for the metric dimension. Also, for every positive integer, we provide a construction showing the existence of a cube of a tree satisfying its metric dimension as the given integer. Further, we characterize some restricted families of cube of trees satisfying $\beta{(T^{3})}=\beta{(T)}$. https://comb-opt.azaruniv.ac.ir/article_14863.html In this article, the edge graceful labeling concept has been expanded from conventional fuzzy graphs to intuitionistic and neutrosophic graphs. There has been much discussion of the edge graceful labeling in intuitionistic and neutrosophic graphs with certain sequence of edge labels(for each membership) in clockwise or anticlockwise direction and the resultant vertices. Also, various irregular properties and application of neutrosophic edge graceful labeling graphs have been discussed in detail. https://comb-opt.azaruniv.ac.ir/article_14915.html In this paper, we begin by introducing the concept of complex Narayana-Lucas sequence. Then we proceed to discuss the concept of harmonic number within the framework of complex Narayana-Lucas sequence. Furthermore, we introduce hybrid numbers in the context of harmonic Narayana-Lucas sequence, accompanied by a set of fundamental definitions and theorems pertaining to these sequence. Additionally, we present several mathematical properties, such as generating functions, Binet formulas, and other significant identities related to these newly introduced sequence. Finally, we also provide source Maple 13 code to verify the occurrence of these newly introduced sequence. https://comb-opt.azaruniv.ac.ir/article_14873.html In this paper, we examine a specific type of random chains and propose a unified approach to studying the degree-based topological indices, including their extreme values. We derive explicit analytical expressions for the expected values and variances of these indices and  we establish the asymptotic behavior of the indices. Specifically, we analyze the first Zagreb index, Sombor index, harmonic index, Geometric-Arithmetic index, Inverse Sum Index, and the second Zagreb index for various general random chains, including random phenylene, random polyphenyl, random hexagonal, and linear chains.  https://comb-opt.azaruniv.ac.ir/article_14917.html The Maker-Breaker domination game is played on a graph $G$ by two players, called Dominator and Staller.  They alternately select an unplayed vertex in $G$. Dominator wins the game if he forms a dominating set while Staller wins the game if she claims all vertices from a closed neighborhood of a vertex.  The game is called D-game if Dominator starts the game and it is an \emph{S-game} when Staller starts the game.  If Dominator is the winner in the D-game (or the S-game), then $\gamma_{MB}(G)$ (or $\gamma_{MB}^{\prime}(G)$) is defined by the minimum number of moves of Dominator to win the game under any strategy of Staller.  Analogously, when Staller is the winner, $\gamma_{SMB}(G)$ and $\gamma_{SMB}^{\prime}(G)$ can be defined in the same way.  We determine the winner of the game on the Cartesian product of paths, stars, and complete bipartite graphs, and how fast the winner wins. We prove that Dominator is the winner on $P_m \square P_n$ in both the D-game and the S-game, and $\gamma_{MB}(P_m \square P_n)$ and $\gamma_{MB}^{\prime}(P_m \square P_n)$ are determined when $m=3$ and $3 \le n \le 5$. Dominator also wins on $G \square H$ in both games if $G$ and $H$ admit nontrivial path covers. Furthermore, we establish the winner in the D-game and the S-game on $K_{m,n} \square K_{m',n'}$ for every positive integers $m, m',n,n'$.  We prove the exact formulas for $\gamma_{MB}(G)$, $\gamma_{MB}^{\prime}(G)$, $\gamma_{SMB}(G)$, and $\gamma_{SMB}^{\prime}(G)$ where $G$ is a product of stars. https://comb-opt.azaruniv.ac.ir/article_14866.html Recently, the Sombor index of a graph has been extended to general Sombor index. The general Sombor index of a simple graph $G$ is defined as $SO_\alpha(G)=\displaystyle\sum_{uv\in E(G)}[d_G(u)^2+d_G(v)^2]^{{\alpha}/2}$, where $d_G(u)$ denotes the degree of a vertex $u$ in $G$ and $\alpha$ is a real number. In this paper, we obtain bounds for the general Sombor index of trees. We further determine the trees with the extremal general Sombor indices. https://comb-opt.azaruniv.ac.ir/article_14918.html In a connected graph $G$, the all-paths transit function $A(u,v)$, consists of the set of all vertices in the graph $G$ which lies on some path connecting $u$ and $v$.  Convexity obtained by the all-paths transit function is called all-paths convexity.  A Radon partition of a set $P$ of vertices of a graph $G$ is a partition of $P$ into two disjoint non-empty subsets such that their convex hulls intersect.  A Radon partition $(P_t, Q_t)$ of $P$ is called $t$-tolerant Radon partition, if for any set $S\subseteq P$ with $|S|\le t$,  the intersection of the convex hulls  $\langle P_t\setminus S \rangle \cap \langle Q_t\setminus S \rangle \neq \emptyset$.  This paper is devoted to $t$-tolerant Radon partitions for the all-paths convexity of connected simple undirected graphs.  It is proved that the minimum number of vertices needed for $t$-tolerant Radon partition is $2t+4$. But, some selection of $2t+4$ vertices of $G$ has a $(t+1)$-tolerant Radon partition.  In this paper, we discuss the necessary and sufficient condition to the existence of $(t+1)$-tolerant Radon partition for $2t+4$ vertices of $G$.  We also develop algorithms to construct the Radon partition, $t$-tolerant Radon partition, and $(t+1)$-tolerant Radon partition of a set of $2t+4$ vertices, if it exists. https://comb-opt.azaruniv.ac.ir/article_14864.html This study focuses on one of the methods for solving a nonlinear multiobjective convex interval-valued variational problem. Namely, the weighting method is used to find its weakly $LU$-efficient solution and $LU$-efficient solution. Therefore, the weighted variational problem is introduced for the given nonlinear multiobjective interval-valued variational problem. Then, under appropriate convexity assumptions, the equivalance between a (weakly) $LU$-efficient solution of the original nonlinear multiobjective interval-valued variational problem and an optimal solution of its associated weighting variational problem is established. https://comb-opt.azaruniv.ac.ir/article_14920.html The exploration of robust bilevel programming problems is a relatively new development in optimization theory. In this study, we examine a bilevel optimization problem in which both the upper-level and lower-level constraints involve uncertainty. By reducing the problem to a single-level, nonlinear and non-smooth program, we explore sufficient optimality conditions and duality theorems for robust optimal solutions of the considered non-smooth uncertain bilevel optimization problem, using Clarke subdifferentials. Leveraging the characteristics of Clarke subdifferentials, we propose Wolfe-type robust dual models. Additionally, we establish various duality theorems, including weak and strong robust duality, in terms of Clarke subdifferentials. Several illustrative examples are presented to confirm the applicability of the results developed. https://comb-opt.azaruniv.ac.ir/article_14876.html In this work, some new properties of the Hankel and Toeplitz matrices are obtained by considering the Mersenne numbers as entries. We developed efficient formulas to compute matrix norms like $\|.\|_1$,  $\|.\|_\infty$, Euclidean norm, spread, and the lower and upper bound for the spectral norm of these matrices. Also, the study shows that these matrices are non-singular for $n=2$ and singular for $n\geq 3$. Furthermore, we presented rank, eigenvalues, principal minors, and the characteristic polynomial of them explicitly. https://comb-opt.azaruniv.ac.ir/article_14922.html In the study of domination in graphs the following Domination Chain of inequalities is well known and well studied: ir(G)≤gamma(G)≤i(G)≤alpha(G)≤Gamma(G)≤(G)≤IR(G), where ir(G) and IR(G) denote the irredundance number and upper irredundance number, gamma(G) and Gamma(G) denote the domination number and upper domination number, and i(G) and alpha(G) denote the independent domination number and vertex independence number of a graph G. The Domination Chain is a consequence of the facts that (i) every maximal independent set is a minimal dominating set and every minimal dominating set is a maximal irredundant set and (ii) the property of being an independent set is hereditary (every subset of an independent set is also an independent set), the property of being a dominating set is superhereditary (every superset of a dominating set is also a dominating set), and the property of being an irredundant set is hereditary. In this paper we consider several other hereditary properties and superhereditary properties which give rise to similar domination chains. https://comb-opt.azaruniv.ac.ir/article_14867.html Let $G=(V, E)$ be a graph with the vertex set $V$ and the edge set $E$. The first Zagreb index of a graph $G$ is defined to be the sum of squares of degrees of all the vertices of the graph. The second Zagreb index of the graph $G$ is the sum of the $d(u)d(v)$ for every edge $uv \in E$, where $d(u)$ and $d(v)$ denote the degree of the vertices $u, v \in V$. In this paper, we propose new lower bounds of the Zagreb indices of unicyclic graphs in terms of the order and the Roman domination number. We prove that $4n-2\left(\gamma_{R}-\left\lceil\dfrac{2n}{3}\right\rceil\right)$ and $4n-3\left(\gamma_{R}-\left\lceil\dfrac{2n}{3}\right\rceil\right)$ are the sharp lower bounds for the first Zagreb index and the second Zagreb index, respectively. Also, we characterize the extremal trees for these lower bounds. https://comb-opt.azaruniv.ac.ir/article_14924.html A {\em weak signed total Italian dominating function} (WSTIDF) of a graph $G$ with vertex set $V(G)$ is defined as afunction $f:V(G)\rightarrow\{-1,1,2\}$ having the property that $\sum_{x\in N(v)}f(x)\ge 1$ for each $v\in V(G)$, where $N(v)$ is theneighborhood of $v$. The weight of a WSTIDF is the sum of its function values over all vertices.The {\em weak signed total Italian domination number} of $G$, denoted by $\gamma_{wstI}(G)$, is the minimum weight of a WSTIDF in $G$.We initiate the study of the weak signed total Italian domination number, and we present different sharp bounds on $\gamma_{wstI}(G)$.In addition, we determine the weak signed total Italian domination number of some classes of graphs. https://comb-opt.azaruniv.ac.ir/article_14874.html In this paper we consider an algorithm for determining a basis for the Terwilliger and quantum adjacency algebras of a distance-regular graph. For the Terwilliger algebra, we consider the generating set. For the quantum adjacency algebra, we consider the generating set consisting of the raising, flat, and lowering matrices. We give optimization method by using generating matrices with a block-matrix structure so that the number of matrix multiplications required is reduced. https://comb-opt.azaruniv.ac.ir/article_14925.html Given a subset $W$ of the vertex set of a graph $G$, the representation multiset $r_m(v|W)$ of a vertex $v$ is the multiset of distances between $v$ and each vertex in $W$. The subset $W$ is called an m-resolving set of $G$ if distinct vertices have distinct representation multisets. Introduced independently by Saenpholphat and Simanjuntak et al., the m-resolving sets of a graph can be used to uniquely identify its vertices. The notion of m-resolving sets has been shown to be equivalent to identification colorings that have been introduced by Chartrand et al. More recently, Kono and Zhang have established that the prism $K_2 \Box C_n$ has an m-resolving set (equivalently, identification coloring) if and only if $n \geq 6$. In this work, we extend their result by determining the multiset dimension of prisms; that is, we determine the minimum cardinality of their m-resolving sets. https://comb-opt.azaruniv.ac.ir/article_14877.html For an ordinary graph $G$, we compute the eigenvalues and the eigenspaces of the signed line graph $\mathcal{L}(\ddot{G})$, where $\ddot{G}$ is obtained from $G$ by inserting a negative parallel edge between every pair of adjacent vertices. As an application, we prove that if $G$ and $H$ share the same vertex degrees, then $\mathcal{L}(\ddot{G})$ and $\mathcal{L}(\ddot{H})$ share the same spectrum. To the best of our knowledge, this construction does not follow the line of any known construction developed for either graphs or signed graphs. Among the other consequences, we emphasize that $\mathcal{L}(\ddot{G})$ is integral (i.e., its spectrum consists entirely of integers), which means that a construction of integral signed graphs has been established simultaneously. https://comb-opt.azaruniv.ac.ir/article_14926.html Consider a graph $\Omega = (\mathcal{V,E})$ that is simple‎, ‎and let $\vartheta_1$ and $\vartheta_2$ be elements of $\mathcal{V}(\Omega)$ suth that $\vartheta_1 \vartheta_2 \in \mathcal{E}(\Omega)$‎. ‎Then‎, ‎$\vartheta_1$ is said to strongly dominate $\vartheta_2$ if $deg\left(\vartheta_1\right) \geq deg \left(\vartheta_2\right)$‎. ‎A set $K$ of $\mathcal{V}(\Omega)$ is identified as a strong dominating set ($sd$-set) if every vertex $\vartheta_2$ outside of $K$ is strongly dominated by at least one node $\vartheta_1$ within $K$‎. ‎The concept of strong domination integrity for $\Omega$ is defined as $\widetilde{SDI}(\Omega) = \mathop{min}_{K \subseteq \mathcal{V}}\{|K|‎ + ‎m(\Omega‎ - ‎K)‎: ‎K$ is a $sd$-set of $\Omega$\}‎. ‎Similarly‎, ‎the set $K \subseteq \mathcal{V} (\Omega)$ is identified as a geodetic dominating set ($gd$-set) if $K$ is both geodetic and dominating set‎. ‎The geodetic domination integrity of $\Omega$ is defined as $\widetilde{GDI} (\Omega) = min \{|K|‎ + ‎m(\Omega‎ - ‎K)‎: ‎K$ is a $gd$-set of $\Omega\}$‎. ‎This paper delves into the study of strong and geodetic domination integrity sets‎, ‎as well as the impact of node removal on these sets‎. ‎Additionally‎, ‎it introduces the concepts of $\widetilde{SDI}$-Excellent and $\widetilde{GDI}$-Excellent graphs‎, ‎provides examples‎, ‎and derives theorems from these graphs. https://comb-opt.azaruniv.ac.ir/article_14888.html A set $B\subseteq V(G)$ is called a $k$-total limited packing set in a graph $G$ if $|B\cap N(v)|\leq k$ for any vertex $v\in V(G)$. The $k$-total limited packing number $L_{k,t}(G)$ is the maximum cardinality of a $k$-total limited packing set in $G$. Here, we give some results on the $k$-total limited packing number of graphs emphasizing trees, especially when $k=2$. We also study the $2$-(total) limited packing number of some product graphs. A $k$-limited packing partition ($k$LPP) of graph $G$ is a partition of $V(G)$ into $k$-limited packing sets. The minimum cardinality of a $k$LPP is called the $k$LPP number of $G$ and is denoted by $\chi_{\times k}(G)$, and we obtain some results for this parameter. https://comb-opt.azaruniv.ac.ir/article_14927.html For a simple connected graph $ G $ with $ V(G)=\{v_{1},v_{2},\dots,v_{n}\} $, let $ d_{ij} $ be the distance between any pair of distinct vertices $ v_{j} $ and $ v_{j}. $ The reciprocal distance Laplacian matrix $ RD^{L}(G) $ of $ G $ is defined by $ RD^{L}(G)=RTr(G)-RD(G) $, where $ RTr(G) $ is the diagonal matrix having $ i $-the entry $ RTr(v_{i})=\sum_{j\in V(G)}\frac{1}{d_{ij}} $ and $ RD(G) $ is the reciprocal distance matrix (also called Harary matrix) having $ (i,j) $-th entry $ \frac{1}{d_{ij}} $ if $ i\neq j $ and zero, otherwise. The set of all $ RD^{L}(G) $-eigenvalues $ \delta_{1}\geq \delta_{2}\geq \dots\geq \delta_{n-1}>\delta_{n} $ is known as the $ RD^{L} $-spectrum (also called reciprocal distance Laplacian spectrum) of $ G $ and $ \delta_{1} $ is called the $ RD^{L}$-spectral radius (also called reciprocal distance Laplacian spectral radius) of $ G. $ We explore various interesting properties of $ RD^{L} $-eigenvalues along with the bounds for $ RD^{L} $-spectral radius. We characterize the corresponding extremal graphs attaining these bounds. https://comb-opt.azaruniv.ac.ir/article_14879.html The topological indices play a crucial role in generating the weighted adjacency matrix, which exhibits significant diversity from both theoretical and application perspectives compared to the ordinary adjacency matrix. One such notable weighted matrix is the geometric-arithmetic matrix, generated from the well-known $GA$ (geometric-arithmetic) index. Here, we focus on a comparative study of the $GA$ index and the geometric-arithmetic energy $\mathcal{GAE}$. We establish several tight bounds on $\mathcal{GAE}$ involving various graph invariants and identify the corresponding extremal graphs. Additionally, we compare the correlation of the molecular property Bp (boiling point) with $GA$ and $\mathcal{GAE}$. Our findings reveal that the Bp shows good correlation with $\mathcal{GAE}$ than with $GA$ index. Furthermore, we examine the role of $\mathcal{GAE}$ in explaining different properties of drugs associated with kidney disease. https://comb-opt.azaruniv.ac.ir/article_14928.html Graph pebbling is a network optimization technique for the movement of resources in transit. A pebbling move in a connected graph $G$ can be defined as a distribution of pebbles on the vertices of a graph, which involves removing two pebbles from a vertex, placing one pebble on one of its adjacent vertices, and discarding the other pebble. For a graph $G$, the pebbling number $f(G)$ is the minimum number of pebbles required such that one pebble is moved to any arbitrary vertex of the graph $G$. Fractals are described as intricate patterns that are identical at different dimensions or identical in all dimensions. In this paper, the strategy of pebbling is applied to Sierpi\'nski graphs which are well known fractals and several critical points are scrutinized and verified for Generalized Sierpi\'nski graph $S(G,t), t \geq 2$, Sierpi\'nski graph $S(K_n, t)$, $t \geq 1$, $n \geq 2$ and Sierpi\'nski triangle graph $S_m$, $m \geq 2$. https://comb-opt.azaruniv.ac.ir/article_14880.html For a graph $G=(V,E)$, a set $S \subseteq V$ is a \textit{dominating set} if every vertex in $V\setminus S$ has a neighbour in $S$.  The \textit{domination number}, denoted by $\gamma(G)$, is the minimum cardinality of a dominating set in $G$ and a dominating set of minimum cardinality is called a \textit{$\gamma(G)$-set}. Cockayne et al. defined a \textit{Roman dominating function} (RDF) on a graph $G = (V,E)$ to be a function $f:V\rightarrow \lbrace 0,1,2\rbrace$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. The \textit{Roman domination number}, denoted by $\gamma_R(G)$, is the minimum weight of an RDF in $G$. An RDF of weight $\gamma_R(G)$ is called a \textit{$\gamma_R(G)$-function}. Eunjeong Yi introduced the \textit{domination value of $v$}, denoted by $DV_G(v)$, to be the number of $\gamma(G)$-sets to which $v$ belongs. In this paper, we extend the idea of domination value to Roman domination. For a vertex $v \in V$, we define the \textit{Roman domination value}, denoted by $R_G(v)$,  as $ R_G(v) = \sum_{f \in \mathcal{F}} f(v)$, where $\mathcal{F}$ denote the set of  all $\gamma_R(G)$-functions.  We also study some basic properties of Roman domination value of vertices for a given graph and determine the Roman domination value for the  vertices of a complete $k$-partite graph. https://comb-opt.azaruniv.ac.ir/article_14931.html In this paper, we obtain a formula for the Harary index and hyper-Wiener index of Mycielskian of $G$, $\mu(G)$, and complement of $\mu(G)$. More precisely, we determine a formula for the hyper-Wiener index of $\mu(G)$ in terms of Zagreb indices of $G$ if the girth of $G$ is greater than $6$ and we deduce the result in [M. Azari in Discrete Math. Algorithms and Appl. 09 (2017) 1750022]. In addition, we find a formula for the vertex Padmakar-Ivan index of $\mu(G)$ if the girth of $G$ is greater than 7 and the complement of $\mu(G)$. https://comb-opt.azaruniv.ac.ir/article_14881.html Let $G=(V,E)$ be a graph. A strong global distribution center of $G$ is a dominating set  $S\subseteq V$  such that for any $v\in V\setminus S$, there exists a vertex $u\in N[v]\cap S$ with the property $|N[u]\cap S|> |N[v]\cap (V\setminus S)|$. The strong global distribution center number, gdc$^s(G)$, of a graph $G$ is the minimum cardinality of a strong global distribution center of $G$. In this paper, we introduce the concept of strong global distribution center. We give some bounds on the gdc$^s(G)$ for general graphs and classify graphs with extremal values of gdc$^s(G)$. Also, we compute the strong global distribution center number for some families of graphs and  study this parameter for some families of graph products. https://comb-opt.azaruniv.ac.ir/article_14934.html In this paper, we introduce the concepts of bilevel vector variational inequalities (BVVI) of both Minty and Stampacchia types. Additionally, we establish connections between BVVI and multiobjective bilevel optimization problems (MBOP), focusing on the use of tangential subdifferentials. We investigate the relationship between the vector efficient points of MBOP and the solutions of BVVI, particularly under conditions of generalized convexity. https://comb-opt.azaruniv.ac.ir/article_15076.html This paper extends Hermite-Hadamard type inequalities within the framework of stochastic fractional calculus. We investigate how fractional integrals, which account for memory effects, interact with random processes. Our work presents three main contributions. First, we provide an error bound for approximating a standard integral of a smooth, deterministic function using stochastic fractional integrals. Second, we extend the well-known Hermite-Hadamard inequality, which applies to convex functions, to the setting of convex stochastic processes, showing how their expected values are bounded by these integrals. Finally, we derive specific mean-square error bounds when approximating a standard Brownian motion using its stochastic fractional integrals. These results enhance our understanding of stochastic fractional inequalities, offering new tools for analyzing complex systems influenced by both memory and randomness. https://comb-opt.azaruniv.ac.ir/article_14937.html The paired, total, and independent domination subdivision number of a graph $G$     is the minimum number of edges that must be subdivided, where each edge can be subdivided at most once, in order to increase the paired, total, and independent domination number, respectively. In this paper,     we prove that the corresponding decision problems for paired, total, and independent domination subdivision numbers are NP-hard, even when restricted to bipartite graphs. Additionally, we point out the error in the previous proof of $\mathrm{NP}$-hardness of the paired domination subdivision problem by Amjadi and Chellali  in  "Complexity of the paired domination subdivision problem" [Commun. Comb. Optim. 7 (2022), No.2, 177–182]. https://comb-opt.azaruniv.ac.ir/article_14938.html In this work, two class NP-hard optimization problems on the graph are discussed: the detour metric dimension and the bi-metric dimension. Both are used in many distinct areas, as well as pattern recognition, keeping track of the movement of robots on a network, and reviewing the structural properties of chemical structures. The metric dimension $dim(G)$ of graph $G$ is the minimum number of vertices such that every vertex of $G$ is uniquely assigned by its vector of distances to the selected vertices. This concept was expanded into the detour metric dimension $D\beta(G)$ and the bi-metric dimension $\beta_{b}(G)$ by considering the detour distance of two vertices. A computational approach is needed to solve these two problems on large graphs. In this research, we propose the BGWO algorithm to determine the metric dimension of some generalized antiprism graphs. In addition, we develop a probabilistic-based metaheuristic algorithm, namely ant colony optimization, to find the detour distance and then modify the binary gray wolf optimization (BGWO) algorithm to solve the detour metric dimension and the bi-metric dimension on some families of graphs. The simulation shows that the BGWO algorithm gives better results for the generalized antiprism graphs. Also, the hybrid ACO-BGWO algorithm gives the same detour dimension result as in the literature. We show that the bi-metric dimension of the generalized antiprism graph is the same as its metric dimension. https://comb-opt.azaruniv.ac.ir/article_14940.html The sign of a vertex in a signed graph be defined naturally as the product of signs of edges incident to the vertex. We say that an edge is {\it consistent} or a  $c$-edge if its end-vertices have the same sign. Over the years, different notions of vertex coloring have been defined for signed graphs. Here, we introduce a new type of coloring in which any two vertices joined by a $c$-edge are assigned different colors. We call this the $s$-coloring of a signed graph. The $s$-chromatic number $\chi _{s}(G)$ of a signed graph $G$ is the minimum number of colors required to properly $s$-color the vertices of $G$. We obtain several bounds for $\chi_{s}(G)$. We show that the number of $s$-colorings of a signed graph $G$ is a polynomial function of the number $k$ of colors, which we call the $s$-chromatic polynomial $S(G,k)$ of $G$. We define the operations of removal and compression to develop a deletion-contraction type recursive procedure for determining $S(G,k)$. We introduce the notions of $c$-complete and $c$-full signed graphs, characterizing different classes of $c$-full signed graphs and determining the number of $c$-complete signed graphs on a given number of vertices. Furthermore, the relationship between $s$-coloring and other signed graph colorings is also investigated. https://comb-opt.azaruniv.ac.ir/article_14941.html A $(p,m,n)$ signed graph $S$, is a signed graph of order $p$ with $m$ positive edges and $n$ negative edges. In this paper, we first prove a few basic results on vertex labelings of paths. We use these results and a sequence of lemmas to obtain a characterization of additively graceful signed paths. We prove that, apart from exactly 4 exceptions, additively graceful signed paths are characterized by the signed paths containing at most one negative section with $n \leq 2$. We also establish a characterization of additively graceful signed cycles. We prove that a $(p,m,n)$ signed cycle $S$ is additively graceful if and only if one among the following 4 conditions are satisfied, (a) $n=0$ and $ m\equiv 0$ or $3 \pmod 4$,  (b) $n=1$ and $ m\equiv 1$ or $2 \pmod 4$,  (c) $n=2$, $ m\equiv 1$ or $2 \pmod 4$ and $S$ contains a single negative section,  (d) $S$ is the all negative signed cycle on $C_3$.  https://comb-opt.azaruniv.ac.ir/article_14942.html An exact doubly dominating set (also called an efficient doubly dominating set in [F. Harary and T.W. Haynes, Double domination in graphs, Ars Combin. 55 (2000), 201–213]) for a graph $G=(V,E)$ is a subset $D$ of vertices such that each vertex of $G$ is dominated by exactly two vertices of $D$. In this paper we  show that subdivision graphs admit exact doubly dominating sets under specific conditions, while Mycielskian and middle graphs do not. We provide some characterizations and we investigate  the existence of exact doubly dominating sets  for their complements. https://comb-opt.azaruniv.ac.ir/article_14943.html Given a commutative ring $R$, a left $R$-module $M$, and an $R$-submodule $N \subseteq  M$, the graph $G(R;M,N)$, induced on $R$ by the pair $(M,N)$, is a simple graph with vertex set $R^* = R \backslash \{0\}$. Distinct vertices r and s are adjacent if $rsN = 0$. This graph generalizes Beck's zero-divisor graph $G(R)$. We analyze connectivity, completeness, bipartiteness, cycles, diameter, girth, independence/clique/chromatic numbers, and domination numbers, often under specific algebraic constraints on $R$ or $N$. Applications to $\mathbb Z_n$-modules illustrate these results. By linking $G(R;M,N)$ to $G(R)$, we derive graph invariants for $G(R)$ efficiently and vice versa, deepening insights into algebraic structures and their graph-theoretic analogs. https://comb-opt.azaruniv.ac.ir/article_14944.html In the ever-evolving landscape of parallel computing architectures, the demand for innovative interconnection networks is paramount. This paper introduces Optical Transpose Interconnection System (OTIS) - Bijective connection graphs, a subclass of interconnection network designed to address the challenges on scalability, efficiency, and fault tolerance. By merging the strengths of networks, namely, OTIS networks and Bijective connection graphs (BC graphs in brief), we aim to overcome the limitations inherent in individual architectures. This paper presents a comprehensive analysis of Optical Transpose Interconnection System - Bijective connection graphs. We demonstrate superiority over traditional interconnection networks, showcasing their potential to emerge as an interesting candidate for parallel computing.Precisely, in this work, we compute few basic graph theoretical parameters, explored the embedding properties, solved the edge isoperimetric problem, and many associated properties of the proposed class of network. https://comb-opt.azaruniv.ac.ir/article_14945.html ‎We consider two generalizations of the Sombor index, obtained by weighting the standard contribution $\sqrt{d_\Omega(\eta)^2+d_\Omega(\eta')^2}$ of an edge $\eta\eta'$ of a graph $\Omega$ by the sum and by the product of degrees of its end-vertices, respectively. The first generalization has been considered under the name of the elliptic Sombor index, while the second one seems to be new. We consider trees and unicyclic graphs on a given number of vertices $n$ with a given maximum degree $\Delta $ and characterize the graphs minimizing both generalizations over those classes of graphs. https://comb-opt.azaruniv.ac.ir/article_14946.html The general Randi\'{c} index of a graph $G$ is defined as $R_{a} (G) = \sum_{uv \in E(G)} [d_G (u) d_G (v)]^{a}$, where $a \in \mathbb{R}$, $E(G)$ is the set of edges of $G$, and $d_G (u)$ and $d_G (v)$ are the degrees of vertices $u$ and $v$, respectively. Among unicyclic graphs with given number of vertices and maximum degree, we present the graph with the largest value of $R_{a}$ for $a < 0$, and graphs having the smallest values of $R_{a}$ for $a > 0$. https://comb-opt.azaruniv.ac.ir/article_14947.html A subset \( S \) of the vertices of a graph is called a \( k \)-dominating set if every vertex outside \( S \) has at least \( k \) neighbors in \( S \). If a \( k \)-dominating set is an independent subset of the vertices, then the set is called an independent \( k \)-dominating set. The size of the smallest such set is called the independent \( k \)-domination number of the graph. In this paper, we derive a lower bound on the independent \( k \)-domination number of Hamming graphs. For some sets of parameters, we show that this lower bound is exact. https://comb-opt.azaruniv.ac.ir/article_14948.html Ptolemaic graphs are precisely the graphs that are both chordal and distance-hereditary. Markenzon et al.~\cite{markenzon} established a hierarchy of Ptolemaic graphs comprising six subfamilies: laminar chordal graphs, block duplicate graphs, block graphs, AC graphs, trees, and paths. In this paper, we present a new proof of the characterization of AC graphs using forbidden induced subgraphs and identify an additional graph class that lies between AC graphs and paths within this hierarchy. The interval function is a well-studied tool in metric graph theory, and the characterization of the interval function of graph families is an interesting problem in metric graph theory having connections to first-order logic. In this paper, we propose a set of independent betweenness axioms for an arbitrary function known as a transit function and provide a characterization of the interval functions corresponding to graphs in the extended hierarchy of subgraphs of Ptolemaic graphs, specifically laminar chordal graphs, block duplicate graphs, and AC graphs. https://comb-opt.azaruniv.ac.ir/article_14949.html A \textit{signed graph} $\Sigma$ is an ordered pair ($\Sigma^{u}$,$\sigma$), where $\Sigma^{u}$=(V,E) is the \textit{underlying graph} and $\sigma$ is sign mapping called \textit{signature}, which assigns each edge in E a sign from the set $\lbrace +, - \rbrace$. The study of vertex-degree-based topological index: known as \textit{Sombor index} was initiated by I. Gutman in 2021 for any graph $G$. He defined it as $SO(G)= \sum_{e_{ij} \in E(G)} \sqrt{d_{G}(v_{i})^2 + d_{G}(v_{j})^2}$. In this work, the concept of the Sombor index is extended to connected signed graphs. The Sombor index is derived mathematically for signed paths and signed cycles, and is supported by computational algorithms. Furthermore, it is proved that the Sombor index of a connected signed graph $\Sigma$ is maximized if and only if the \textit{net-degree variance} of $\Sigma$ is also maximized. As an application, this study provides a solution to the net-degree variance maximization problem for certain types of signed graphs.\\ https://comb-opt.azaruniv.ac.ir/article_14953.html For a given graph $G$, let $\theta_f(G)$ denote the minimum number of stars (not necessarily induced) needed to cover the vertices of $G$, and let $\alpha_f(G)$ denote the maximum number of vertices in a set $S \subseteq V(G)$ such that no two distinct vertices $u, v \in S$ belong to the same subgraph of $G$ that is a star. Clearly, $\theta_f(G) \geq \alpha_f(G)$. A graph $G$ is said to be \emph{non-induced star-perfect} if $\theta_f(H) = \alpha_f(H)$ for every induced subgraph $H$ of $G$. A graph $G$ is a \emph{domination graph} if every induced subgraph $H$ of $G$ contains a pair of vertices $x, y$ such that $N_H(x) \subseteq N_H[y]$. In this paper, we investigate domination graphs that are non-induced star-perfect and explore well-known subclasses within this category. Additionally, we present an integer linear programming formulation that characterizes a polytope associated with the star-covering set and star-independence set of a graph. https://comb-opt.azaruniv.ac.ir/article_14955.html The commuting graph ${\Gamma(G)}$ of a group $G$ is the simple undirected graph with group elements as a vertex set and two elements $x$ and $y$ are adjacent if and only if $xy=yx$ in $G$. By eliminating the identity element of $G$ and all the dominant vertices of $\Gamma(G)$, the resulting subgraphs of $\Gamma(G)$ are $\Gamma^*(G)$ and $\Gamma^{**}(G)$, respectively. In this paper, we classify all the finite groups $G$ such that the graph $\Delta(G) \in \{\Gamma(G), \Gamma^*(G), \Gamma^{**}(G)\}$ is the line graph of some graph. We also classify all the finite groups $G$ whose graph $\Delta(G) \in \{\Gamma(G), \Gamma^*(G), \Gamma^{**}(G)\}$ is the complement of line graph. https://comb-opt.azaruniv.ac.ir/article_14962.html A generalized subdivision $H'$ of a digraph $H$ is obtained by replacing each arc $e=(x,y)\in E(H)$ with tail $x$ and head $y$, by an oriented path $P_e$ whose first arc has tail $x$ and whose last arc has head $y$, all these new paths being internally disjoint. If all these new paths are directed ones, then $H'$ is simply a subdivision of $H$. The number of blocks (which turns out to have the same parity of $|E(H)|$) of the generalized subdivision $H'$ of $H$ is the sum of all the number of blocks of the new paths $P_e$. In this paper, we prove that if $D$ is spanned by a subdivision of a digraph $H$ such that $\chi(D)$ is at least $2n+|V(H)|+|E(H)|$, then $D$ contains a generalized subdivision of $H$ with $n$ blocks. This bound is simplified when $H$ is an oriented tree. If $H$ is an oriented cycle, then our results assert a special case of a conjecture of Cohen et al. Moreover, the bound is improved to $2n+1$ if $H$ is an oriented cycle with two blocks or $H$ is a directed cycle. https://comb-opt.azaruniv.ac.ir/article_14963.html A derangement $k$-representation of a graph $G$ is a map $\pi$ of $V(G)$ to the symmetric group $S_k$, such that for any two vertices $v$ and $u$ of $V(G)$, $v $ and $u$ are adjacent if and only if $\pi(v)(i) \neq \pi(u)(i)$ for each $i \in \{1,2,3,\ldots,k\}$. The derangement representation number of $G$ denoted by $drn(G)$, is the minimum of $k$ such that $G$ has a derangement $k$-representation. In this paper, we prove that any graph has a derangement $k$-representation. Also, we obtain some lower and upper bounds for $drn(G)$, in terms of the basic parameters of $G$. Finally, we determine the exact value or give the better bounds of the derangement representation number of some classes of graphs. https://comb-opt.azaruniv.ac.ir/article_14965.html The spectral properties of extended adjacency matrices possess high discriminating power and correlate well with various physicochemical properties and biological activities of organic compounds. In the current article, a detailed investigation of one of the extended adjacency matrices called the eccentricity sum matrix is undertaken. The eccentricity sum matrix of a graph G, denoted by A_(ε^c ) (G) is a real symmetric matrix that if i≠ j and v_i v_j∈ E(G), then the (i,j)^th- entry is e(v_i)+e(v_j) and zero otherwise, where e(v_i) is the eccentricity of vertex v_i. The properties like trace, principle minors, and eigenvalues of the eccentricity sum matrix are explored. Moreover, we present some bounds for spectral radius and energy. Also, the energy and spectrum of some classes of graphs like fan graphs, bi-star graphs, etc., and their complements are obtained. https://comb-opt.azaruniv.ac.ir/article_14969.html Among the defined $148$ discrete Adriatic indices, the max-min degree index is one. Vuki{\'c}evi\'c proposed some problems related to the upper and lower bounds on the max-min degree index. Here we determine the max-min degree index of some special graphs. We characterize the graphs extremal with respect to max-min degree index over connected graphs, trees and unicyclic graphs with a given number of vertices. Finally, we establish its mathematical relation with other topological indices. https://comb-opt.azaruniv.ac.ir/article_14970.html According to Nikiforov [V. Nikiforov, Merging the A- and Q-spectral theories, Appl. Anal. Discrete Math. 11 (2017), no. 1, 81–107], the \(A_\alpha\)-matrix of a graph \(G\) is defined as \(A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)\), where \(\alpha \in [0, 1]\), \(D(G)\) is the diagonal matrix with the degrees of the vertices of \(G\) as the diagonal entries, and \(A(G)\) is the adjacency matrix. The \(A_\alpha\)-spectral radius of the \(A_\alpha\)-matrix is its largest eigenvalue.In this study, we characterize the graph that maximizes the \(A_\alpha\)-spectral radius within three specific classes of graphs: (i) graphs of order \(n\), with vertex connectivity \(\kappa(G) \leq k\) and minimum degree \(\delta(G) \geq k\); (ii) bipartite graphs of order \(n\) with vertex connectivity \(k\); and (iii) graphs of order \(n\), connectivity \(k\), and independence number \(r\). Furthermore, for each of these three families, we determine the location of the \(A_\alpha\)-spectral radius. https://comb-opt.azaruniv.ac.ir/article_14971.html Mathematical chemistry is a branch of mathematics that uses mathematical tools to solve chemical problems. In this paper, we calculate the harmonic index and harmonic polynomial of some chemical graphs, among graphene, star dendrimers, polyomino chains from $n$ cycles and triangular benzene, polymolecular graphs, carbon nanotube networks, two classes of benzene-like series, nanocone, spiro chain, and polyphenylenes. https://comb-opt.azaruniv.ac.ir/article_14972.html Let $k\geq 1$ be an integer. A signed total Italian $k$-dominating function (STIkDF) on a graph $G=(V, E)$ is a function $f: V\rightarrow \{-1,1,2\}$ satisfying the conditions that $\sum_{u\in N(v)}f(u)\geq k$ for each vertex $v\in V$, where $N(v)$ is the neighborhood of $v$, and each vertex $u$ with $f(u)=-1$ is adjacent to a vertex $v$ with $f(v)=2$ or to two vertices $w$ and $z$ with $f(w)=f(z)=1$. The weight of an STIkDF $f$ is $w(f)=\sum_{v\in V}f(v)$. The signed total Italian $k$-domination number of $G$, denoted by $\gamma_{stI}^k(G)$, is the minimum weight of an STIkDF on $G$. In this paper, we prove that the decision problem for the signed total $k$-domination is NP-complete for $k\in\{1,2\}$. We present tight lower bound on \textcolor{red}{$\gamma_{stI}^2(G)$}, and characterize all extremal graphs. Using a discharging method, we also determine the value \textcolor{red}{$\gamma_{stI}^2(C_3\Box C_n)$} for all $n\geq 3$. https://comb-opt.azaruniv.ac.ir/article_14973.html A full Nesterov-Todd step interior point method is designed and analyzed in this paper to solve the weighted linear complementarity problem in Euclidean Jordan algebra. Under appropriate conditions, it is proven that the full Nesterov-Todd step is strictly feasible and the algorithm has a quadratic convergence rate to the target point on the central path in the framework of Euclidean Jordan algebras. The obtained iteration bound for the algorithm matches the best known current iteration bound for this problem. To the best of our knowledge, this is the first full-step interior point algorithm for the weighted complementarity problem in the space of Euclidean Jordan algebras. https://comb-opt.azaruniv.ac.ir/article_14979.html Let $(P, \leq)$ be an atomic partially ordered set (briefly, a poset) with a minimum element $0$, and let $\mathcal{I}(P)$ be the set of all nontrivial ideals of $ P $. The essential graph of $P$, denoted by $G_e(P)$, is an undirected, simple graph with the vertex set $\mathcal{I}(P)$ and two distinct vertices $I, J \in \mathcal{I}(P) $ are adjacent in $G_e(P)$ if and only if $ I\cup J $ is an essential ideal of $P$. We study the connections between the graph-theoretic properties of this graph and the algebraic properties of a poset. We prove that $G_e(P)$ is connected with diameter at most three. Furthermore, all posets are characterized based on the diameters of their essential graphs. Also, all posets with planar $G_e(P)$ are classified. Among other results, the clique number and chromatic number of $G_e(P)$ are determined. https://comb-opt.azaruniv.ac.ir/article_14980.html For an arbitrary invariant $\rho(G)$ of a graph $G$ the $\rho$-edge adding stability number $eas_{\rho}(G)$ is the minimum number of edges of the complement $\overline{G}$ whose addition to $G$ results in a graph $H \supseteq G$ with $\rho(H) \neq \rho(G)$. If such an edge set does not exist, then we set $eas_{\rho}(G) = \infty$. In the first part of this paper we give some general results for $eas_{\rho}(G)$. We prove among others a Gallai's theorem type result for invariants that are based on the $\rho$-edge adding stability number. https://comb-opt.azaruniv.ac.ir/article_14981.html We study structural properties of cyclic codes, and their generalization, over a general infinite family of rings, namely the ring $\mathcal{R}_k$ defined by $R[v_1,v_2,\ldots,v_k]$ with conditions $v_i^2=v_i,$ for $i \in [1,k]_\ZZ,$ where $R$ is any finite commutative Frobenius ring. We derived necessary and sufficient condition for the codes to be cyclic, quasi-cyclic, skew-cyclic as well as to be quasi-skew-cyclic. As an application, we constructed optimal linear codes over $\ZZ_4$ as a Gray images of our codes. https://comb-opt.azaruniv.ac.ir/article_14982.html Topological indices are descriptors that assign a number to each molecular graph, often well correlated to some properties. In particular, the Schultz index has stood out for its high discrimination capacity between different molecular structures, being a key tool in the study of their physicochemical properties. In this paper, we introduce a modification of the classical adjacency matrix making use of the Schultz index, incorporating both the degree of the vertices and the distance between each pair of them. We perform a spectral analysis of this index and identify some of its significant properties. Particularly, we focus on determining upper and lower bounds for the eigenvalues of this matrix, contributing to the understanding of its algebraic structure and its relationship with graph parameters.  https://comb-opt.azaruniv.ac.ir/article_14983.html A signed graph $\Sigma=(G,\sigma)$ is a graph $G$ together with a signature function $\sigma$ which assigns $1$ or $-1$ on the edges of $G$. Seidel matrix of an unsigned graph is already defined and researchers investigated some of its spectral and other properties. Considering the recently introduced notion of signed distance in signed graphs and that of the distance compatible signed graphs, we define distance induced Seidel matrices for such signed graphs and analyze their spectrum mainly for some classes of unbalanced distance compatible signed graphs, as balanced signed graphs possess the same distance induced Seidel spectrum as that of its underlying graph. We also deal with the distance compatibility issue in the line graph of a distance compatible signed graph and discuss the corresponding distance induced Seidel spectrum in this regard. https://comb-opt.azaruniv.ac.ir/article_14984.html The aim of this article is to develop necessary and sufficient optimality conditions for nonsmooth mathematical programs with equilibrium constraints $(\mathcal{MPEC})$. We introduce a nonsmooth variant of the standard $\partial^{T}$-Abadie constraint qualification ($\partial^{T}$-$ACQ(\mathfrak{B}_1, \mathfrak{B}_2)$) and propose $\partial^{T}$-generalized alternatively stationary conditions using the tangential subdifferential framework. Building on these new conditions, we derive first-order optimality criteria under $\partial^{T}$-$ACQ(\mathfrak{B}_1, \mathfrak{B}_2)$. Additionally, we establish sufficient optimality conditions within a framework of generalized convexity assumptions. The effectiveness and applicability of these conditions are demonstrated through several examples. https://comb-opt.azaruniv.ac.ir/article_14985.html The minimum lateness scheduling problem seeks to create a schedule that minimizes the largest lateness of the system. This paper deals with the challenge of increasing the processing time of jobs in a minimum cost such that the minimum lateness attains a given bound. It is called the minimum cost problem of downgrading minimum lateness scheduling. Additionally, the modifying costs are represented as intervals, and we apply the minmax regret criterion to address this uncertainty. Our contribution is an $O(n^2)$ algorithm for solving the corresponding robust problem. https://comb-opt.azaruniv.ac.ir/article_14986.html Identifying master regulatory genes is crucial for analyzing gene regulatory networks. Various optimization-based approaches have been developed to identify potential sets of master regulatory genes. In a weighted gene regulatory network, each interaction between gene pairs is assigned a weight. In such networks, not only  direct interactions between genes significant, but indirect influences also play an important role. In this study, an indirect relationship between two genes is considered to exist when, in addition to a potential direct link, there is at least one additional pathway through which they influence each other. An influence value between two genes is calculated using an algorithm inspired by the $K$-shortest path approach. Furthermore, each gene is assigned an impact factor based on its overall influence within the weighted network. These tools allow us to introduce a new method based on a modified version of the well-known uncapacitated facility location problem. This method can identify the most significant genes among those detected by other approaches and also determine a master regulatory gene that controls a specific target gene. The proposed approach has been applied to several gene regulatory networks, and the results are reported and compared against two existing models. https://comb-opt.azaruniv.ac.ir/article_14990.html The family of mathematical objects known as tournaments constitutes one of the main areas of directed graphs, being defined as having a set of vertices in which each pair is joined by exactly one arc. Here we are interested in a secondary family known as bipartite tournaments. These are defined as having two non-empty sets of vertices and one arc joining each pair that are in different sets. There is a substantial theory of this topic also, and this article presents many of its foundational areas. One key concept is that of the number of arcs going from a vertex, called its score, and the theory involves what lists of numbers can constitute the scores. The cycles in a bipartite tournament have a variety of interesting collections involving lengths. Other fundamental concepts that we discuss include bipartite tournaments with the property of being self-converse, the efficient removal of cycles by the reversal of arcs, development of a theory of upsets, and properties of automorphism groups of the structures. The concept of bipartite tournaments also generalizes naturally to multipartite tournaments with more parts. Many of the results we discuss in this paper lead to questions about analogous extensions to these structures. In the last section of the paper, we include a discussion of further research directions. https://comb-opt.azaruniv.ac.ir/article_14991.html In this paper, we present a novel perspective on vertex-degree-based topological indices. Established degree--based topological indices are based on adjacent vertices. One could contemplate including all pairs of vertices. Recently, Gutman introduced the Sombor indices. Here, we introduce the extended versions of the Sombor indices including all pairs of vertices in the Sombor indices formula. We explore the fundamental mathematical properties of these extended indices, establish upper and lower bounds in terms of some graph parameters, and find the sharp bounds. Additionally, we determine the extremal chemical trees with maximum and minimum extended Sombor index. Moreover, the role of extended Sombor indices in describing structure–property relationships is demonstrated. https://comb-opt.azaruniv.ac.ir/article_14992.html In this article, we investigate the solution of constrained optimization problems using the Karush Kuhn Tucker (KKT) condition with single-valued neutrosophic triangular number coefficients. Our approach introduces new neutrosophic arithmetic operations applied to the parametric representations of neutrosophic numbers, along with the neutrosophic ranking of the parametric forms of Triangular Neutrosophic Numbers. The primary objective of this study is to develop a robust framework for solving constrained Single-Valued Neutrosophic Nonlinear Programming Problems using the KKT condition, effectively managing uncertainty and imprecision in optimization. We present and prove an important theorem for the KKT condition under neutrosophic environments, contributing to the theoretical foundation of this method. Furthermore, a detailed numerical example illustrates the practical application of the proposed approach. The results are compared with those of existing methods, demonstrating the effectiveness and advantages of the neutrosophic-based solution. https://comb-opt.azaruniv.ac.ir/article_14993.html Given a nontrivial connected undirected graph $G$ with diameter $d$, a vertex coloring $c$ of $G$ that uses only the colors red and white induces, for each $v \in V(G)$, the $d$-vector $\vec{d}(v) = [a_1 a_2 \cdots a_d]$, where each $a_i$ is equal to the number of red vertices of distance $i$ from $v$. Then $c$ is called an ID-coloring of $G$ if $\vec{d}(v) \neq \vec{d}(w)$ for all distinct $v,w \in V(G)$. If $G$ has at least one ID-coloring, then it is called an ID-graph and its identification number $ID(G)$ is defined to be the minimum number of red vertices among all ID-colorings of $G$. The notions of ID-colorings and identification number have been shown to be equivalent to the notions of multiset resolving sets and multiset dimension, respectively. Previous works on this topic have focused on characterizing ID-caterpillars and ID-lobsters and on the identification numbers of some ID-caterpillars. In this paper, we focus on the identification numbers of ID-lobsters. Specifically, we establish a sharp lower bound for the identification number of all ID-lobsters. Furthermore, we characterize and determine the identification numbers of all uniform ID-lobsters. https://comb-opt.azaruniv.ac.ir/article_14994.html The study of atom-bond sum-connectivity index emerged recently as a variant of atom-bond connectivity index by replacing the product in the denominator in each of the fractions corresponding to every edge by the sum of the degrees. In this paper, we established relationships between the $ABS$ index with some other existing degree-based topological indices in terms of minimum degree and maximum degree of the graph. https://comb-opt.azaruniv.ac.ir/article_14995.html Let $G$ be a graph and $s$ be an integer. {\textit{A $s$-container $C(x,y)$} of $G$ between two vertices $x$ and $y$ is a set of $s$ internally vertex disjoint $x,y$-paths. A $s$-container $C(x,y)$ is \textit{a $s^{*}$-container} if $V(C(x,y))=V(G)$, where $V(C(x, y))$ is the set of vertices incident with some paths in $C(x,y)$. Then $G$ is \textit{$s^{*}$-connected} if there exists a $s^{*}$-container between any two distinct vertices of $G$. \textit{The spanning connectivity $\kappa^{*}(G)$} of $G$ is the largest integer $k$ such that $G$ is $s^{*}$-connected for any $s$ with $1 \leq s \leq k$. Further, $G$ is \textit{super spanning connected} if $\kappa^{*}(G)=\kappa(G)$, where $\kappa(G)$ is the connectivity of $G$. In this paper, we show that the $n$-th cartesian product of complete graph $K_{t}$ $(t\ge 3)$ is super spanning connected. Our results, in some sense, extended a previous result in \textit{[Shih et al., One-to-one disjoint path covers on $k$-ary $n$-cubes, Theoret. Comput. Sci. (2011)]}. https://comb-opt.azaruniv.ac.ir/article_15000.html This note contains a new combinatorial proof of Cramer’s rule based onthe Gessel-Viennot-Lindström Lemma. https://comb-opt.azaruniv.ac.ir/article_15001.html This paper determines the proper $D$-Lucky edge numbers for human chain graphs, circular human chain graphs, strong human chain graphs, and weak human chain graphs. https://comb-opt.azaruniv.ac.ir/article_15002.html In this paper, we study a quadratic minimization problem over the intersection of a ball and a reverse ball constraints that includes generalized trust-region subproblem (TRS). Using the structure of the problem, we prove that it can be solved to global optimality by solving at most three TRS or two TRS with an extra linear constraint. Then we present an efficient TRS-based algorithm to solve it. Computational experiments illustrate that our new algorithm outperforms the ones in the literature, specially the algorithm for generalized TRS, on three widely used test classes. https://comb-opt.azaruniv.ac.ir/article_15003.html The elliptic Sombor index is a topological index based on vertex degree introduced by  Gutman. Suppose $G=(V(G), E(G))$ is a finite, connected, and simple graph with  $V(G)=\{w_1, w_2, \dots, w_p\}$. Suppose $d_{G}(w_i)$ isthe degree of  $w_i$, for $1\leq i \leq p$. We use $ES(G)$ to represent the Sombor elliptic matrix $G$ which is  a$p\times p$ matrix   and its $(i, j)$-entry is equal to $(d_{G}(w_{i})+d_{G}(w_{j}))\sqrt{d_{G}^{2}(w_{i})+d_{G}^{2}(w_{j})}$ if $w_{i}w_{j}\in E(G)$, and zero otherwise. We introduce and investigate the elliptic Sombor energy and elliptic SomborEstrada index, both base on the eigenvalues of the elliptic Sombor matrix. In addition, we prove some bounds for these new graph invariants. https://comb-opt.azaruniv.ac.ir/article_15004.html ‎For a given integer $k \ge 1$‎, ‎a subset $S$ of vertices of a graph $G$ is a $k$-limited packing if $|N_G[v] \cap S| \le k$ for all $v\in V(G)$‎, ‎where $N_G[v]$ denotes the closed neighborhood of a vertex~$v$ in $G$‎. ‎The $k$-limited packing number‎, ‎$L_k(G)$‎, ‎is the maximum cardinality of a $k$-limited packing in $G$‎. ‎In this paper we present a probabilistic lower bound for the $k$-limited packing number of a graph‎. ‎In particular we improve a previous lower bound given in [Discrete Appl‎. ‎Math‎. ‎184 (2015)‎, ‎146--153]‎. ‎We also present a randomized algorithm for the $k$-limited packing number of a graph‎. https://comb-opt.azaruniv.ac.ir/article_15005.html The energy of a graph G is determined by the absolute sum of its eigenvalues. Similar to this concept, the distance energy, Harary Energy, Seidel energy, complementary distance energy and reciprocal complementary distance energy are all defined based on the eigenvalues of their respective matrices. In this paper, we study these energies on the complement of a regular graph G in terms of the energy of G. We explore exact relationships among these energies. Recent studies have explored equienergetic graphs concerning the adjacency and distance matrices. In this paper, we provide graphs illustrating the equienergetic properties with respect to six matrices. The results obtained extend some of the existing findings. https://comb-opt.azaruniv.ac.ir/article_15006.html The study of topological descriptors is very beneficial in determining the underlying topologies of graphs and networks.An extensive collection of graph-associated numerical descriptors has been used to examine the whole structure of networks. In this analysis, eccentricity-based topological indices have secured a significant place in theoretical chemistry and nanotechnology. Also, graph products conveniently play an essential role in many combinatorial applications, graph decompositions, pure mathematics, and applied mathematics. In this article, we derive the precise results for the eccentric adjacency index of some graph products such as composition, Indu-Bala, Cartesian, disjunction, and symmetric difference products. Furthermore, we implement these outcomes to deduce the eccentric adjacency index for certain significant classes of chemical structures in the factors of graph products. The chemical significance of the index is also investigated. https://comb-opt.azaruniv.ac.ir/article_15007.html The neighbourhood corona $G \star H$ of two graphs $G$ and $H$ is obtained by taking one copy of $G$ and $|V(G)|$ copies of $H$ and making all the neighbours of the $i^{\text{th}}$ vertex of $G$ adjacent with all the vertices in the $i^{\text{th}}$ copy of $H$. In this paper we describe the distance eigenvalues and corresponding eigenvectors of $G \star H$ in terms of the adjacency spectrum of $G$ and $H$ when $G$ is a regular triangle-free graph with diameter 2 and $H$ is regular. Several constructions are proposed using line graphs, iterated line graphs, double graphs, strong double graphs and complement graphs to obtain infinitely many distance non-cospectral pairs of distance equienergetic graphs and non-isomorphic pairs of distance cospectral graphs. Also we obtain the distance Laplacian spectrum of $G \star H$ in terms of the distance Laplacian spectrum of $G$ and Laplacian spectrum of $H$ when $G$ is a transmission regular triangle-free graph with diameter 2. Further we find the distance signless Laplacian spectrum of $G \star H$ in terms of the distance signless Laplacian spectrum of $G$ and signless Laplacian spectrum of $H$ when $G$ is a transmission regular triangle-free graph with diameter 2 and $H$ is regular. We also construct infinitely many non-isomorphic pairs of distance Laplacian cospectral graphs and distance signless Laplacian cospectral graphs. https://comb-opt.azaruniv.ac.ir/article_15008.html Let $k\ge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$. Volkmann \cite{vo21} defined the signed total Italian $k$-dominating function (STIkDF) on a graph $G$ as a function $f:V(G)\rightarrow\{-1,1,2\}$ satisfying the conditions that $\sum_{x\in N(v)}f(x)\ge k$ for each vertex $v\in V(G)$, where $N(v)$ is the neighborhood of $v$, and every vertex $u$ for which $f(u)=-1$ is adjacent to at least one vertex $v$ for which $f(v)=2$ or adjacent to two vertices $w$ and $z$ with $f(w)=f(z)=1$. The weight of an STIkDF $f$ is $w(f)=\sum_{v\in V(G)}f(v)$. The signed total Italian $k$-domination number $\gamma_{stI}^k(G)$ of $G$ is the minimum weight of an STIkDF on $G$. In this paper we continue the study of the signed total Italian $k$-domination number. We present new bounds on $\gamma_{stI}^k(G)$, and we determine the signed total Italian $k$-domination number of some complete $p$-partite graphs. Furthermore, we show that the difference $\gamma_{stR}^k(G)-\gamma_{stI}^k(G)$ can be arbitrarily large, where $\gamma_{stR}^k(G)$ is the signed total Roman $k$-domination number. https://comb-opt.azaruniv.ac.ir/article_15009.html Hypergraphs generalize traditional graphs by allowing edges to connect more than two vertices, enabling a richer representation of relationships in complex systems. Forgotten topological index, or simply $F$-index of a hypergraph, is defined as the sum of cubes of the degrees of all the vertices of the hypergraph. Initially, some sharp bounds for the $F$-index of hypergraphs in terms of other degree-based topological indices have been obtained. A minimally connected hypergraph is a connected hypergraph such that the removal of any hyperedge disconnects the hypergraph. We have characterized the extremal minimally connected hypergraphs corresponding to the $F$-index among  minimally connected hypergraphs on $n$ vertices. The hyperstar and hyperpath with minimum and maximum $F$-indices have been studied. The upper and lower bounds for the $F$-index of the hypergraphs and bipartite hypergraphs are also given. We conclude this article by computing the $F$-index of join, corona product, and Cartesian product of two hypergraphs. https://comb-opt.azaruniv.ac.ir/article_15010.html Let $G$ be an arbitrary simple connected graph. In this paper, we introduce Harary-Sombor index of $G$ and denote it by $HSO(G)$. Then we calculate its values for several familiar classes of graphs. Also, we state an upper bound for the Harary-Sombor index of bipartite graphs. Moreover, we determine the extremum values of the Harary-Sombor index of trees. https://comb-opt.azaruniv.ac.ir/article_15011.html In this article, some upper and lower bounds for the first and the second Zagreb indices of an arbitrary tree in terms of its order and double Roman domination number $\gamma_{dR}$, (depending on whether $\gamma_{dR}$ is odd or even), are stated. Also, all extremal trees attaining equality are characterized. https://comb-opt.azaruniv.ac.ir/article_15012.html The aim of this paper is to provide some operators for symmetric functions for the purpose of obtaining new generating functions for products of k-Fibonacci, $K$-Lucas and $k$-Jacobsthal numbers, $K$-Mersenne numbers bivariate complex Fibonacci poly nomials and Cheb yshev poly nomials with bivariate Mersenne and bivariate Mersenne Lucas polynomials . https://comb-opt.azaruniv.ac.ir/article_15013.html It is conjectured that the crossing number of the complete bipartite graph $K_{m,n}$ without one edge $e$ is equal to $\big \lfloor \frac{m}{2} \big \rfloor \big \lfloor \frac{m-1}{2} \big \rfloor \big \lfloor \frac{n}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor-\big \lfloor \frac{m-1}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor$. In this paper, we establish the validity of this conjecture for $m=5$ using combinatorial properties of cyclic permutations with proofs that can be generalized to all graphs $K_{m,n}\setminus e$ if $m$ is at least six. Further, we give a conjecture concerning crossing numbers of $K_{m,n}$ without several edges incident with a common vertex. https://comb-opt.azaruniv.ac.ir/article_15014.html Many numerical procedures for finding efficient solutions of multiobjective optimization problems are variants of Newton method, that utilize the Hessian matrix of second derivatives. Quasi-Newton methods are used for situations in which the calculation of the Hessian matrix or its inverse is difficult or expensive. In the quasi-Newton methods, only first derivatives are utilized to build an approximation of the actual Hessian matrix over a number of iterations. One of the weaknesses of Newton and quasi-Newton methods is choosing the proper starting points. In fact, the starting points should be close enough to the nondominated solution to have at least quadratic convergence. Therefore, in this study, by applying the convex hull of the individual minimums (CHIMs), we present a procedure for selecting an appropriate starting point for the quasi-Newton method with the BFGS (Broyden, Fletcher, Goldfarb and Shanno) approximation. Moreover, a new algorithm for constructing a uniform approximation of the Pareto front is presented, which can produce more than one efficient point located on the Pareto front in each iteration. To comprehensively compare the proposed algorithm with existing algorithms, three indices are considered: purity metric, measures of coverage, and spacing metric. Extensive numerical experiments show the significant advantage of the proposed algorithm. Moreover, the obtained boundary approximation follows an almost uniform distribution. https://comb-opt.azaruniv.ac.ir/article_15015.html Let $\eta_{1}\ge \eta_{2}\ge\cdots\ge \eta_{n}$ be the eigenvalues of $\mathcal{ABS}$ matrix. In this paper, we characterize connected graphs with $\mathcal{ABS}$ eigenvalue $\eta_{n}>-1$. As a result, we determine all connected graphs with exactly two distinct $\mathcal{ABS}$ eigenvalues. We show that a connected bipartite graph has three distinct $\mathcal{ABS}$ eigenvalues if and only if it is a complete bipartite graph. Furthermore, we present some bounds for the $\mathcal{ABS}$ spectral radius (resp. $\mathcal{ABS}$ energy) and characterize extremal graphs. Also, we obtain a relation between $\mathcal{ABC}$ energy and $\mathcal{ABS}$ energy. Finally, the chemical importance of $\mathcal{ABS}$ energy is investigated and it shown that the $\mathcal{ABS}$ energy is useful in predicting certain properties of molecules. https://comb-opt.azaruniv.ac.ir/article_15016.html In this work, we characterize the class of word-representable graphs with respect to the modular decomposition. Consequently, we determine the representation number of a word-representable graph in terms of the permutation-representation numbers of the subgraphs induced by modules and the representation number of the associated quotient graph. In this context, we also obtain a complete answer to the open problem posed by Kitaev and Lozin on the word-representability of the lexicographical product of graphs. https://comb-opt.azaruniv.ac.ir/article_15020.html Industries are increasingly relying on analytical approaches for performance evaluation and decision-making. Consequently,  they must invest suitable resources at the right time for the appropriate engagements. Inverse  Data Envelopment Analysis is a post-DEA sensitivity analysis method designed to tackle resource allocation. The primary objective of Inverse DEA is to determine the optimal input and/or output levels for each decision-making unit under varying conditions to achieve a specified efficiency target. Traditional inverse DEA models require precise data on the inputs and outputs of Decision-Making Units. However,  in many scenarios,  such as system flexibility,  social and cultural contexts information may be indeterminate. In these cases,  experts' opinions are used to model uncertainty. Uncertainty theory,  a branch of mathematics,  logically deals with degrees of belief. This paper aims to develop an InvDEA model incorporating uncertainty theory. We assume that inputs and outputs of decision-making units are based on experts' belief degrees. An input-oriented model is developed,  and several properties are proven. To demonstrate the model is performance,  we employ a case study involving  CO$_2$ emission data from OPEC countries. https://comb-opt.azaruniv.ac.ir/article_15021.html In this paper we have developed some algebraic structures for the set Fibonacci matrices over initial value spaces ring and field and shown that set of all Fibonacci matrices forms a ring or field (coined as Fibonacci Ring or Fibonacci Field) in either cases. We also investigated those structures over Z; Q; R and C and found that over Q it forms a Fibonacci Field but over Z; R and C it is a Fibonacci Ring. Finally we have introduced a new concept of f-inverse initial value along with that of f-congruent equivalence class and demonstrated graphically which leads a wide scope of future work. https://comb-opt.azaruniv.ac.ir/article_15026.html A (proper) coloring of $G$ is said to be domination coloring if each vertex dominates at least one color class and each color class dominated by some vertex. The minimum number of colors required for a domination coloring of $G$ is called the domination chromatic number of $G$, and is denoted by $\chi_{dd}(G)$. For the graph $G$ without isolated vertices, domination coloring of $G$ is said to be the total domination coloring if each vertex dominates at least one color class contained in its open neighborhood. The minimum number of colors required for a total domination coloring of $G$ is called the total domination chromatic number, and is denoted by $\chi_{td}(G)$. In this paper, we have constructed graphs $G$ for arbitrary values of  $\chi(G)$ and $\chi_{dd}(G)$, as well as  $\chi(G)$ and $\chi_{td}(G)$. An upper bound for domination chromatic number (total domination chromatic number) in terms of dominated chromatic number and domination number (total domination number) is obtained. An upper bound for domination chromatic number in terms of maximum degree and order, as well as in terms of diameter of the graph is obtained. We have also characterized graphs of order $n$ with $\chi_{dom} = n-1$, $\chi_{dd} = 3$ and $\chi_{td} = 3$.  https://comb-opt.azaruniv.ac.ir/article_15027.html In this paper, we study $\overline{G(\tau)}$, the complement graph of the completely separated topological graph, and its line graph $L(\overline{G(\tau)})$ on a topological space $(X, \tau)$. We show that for a discrete topological space $(X, \tau)$, $\overline{G(\tau)}$ is Hamiltonian and Eulerian if and only if $|X|\geq 3$, and for any topological space $(X, \tau)$ such that $|X|\geq 3$, $e(X\backslash \{p\})=2$ for all $p \in X$ if and only if $(X,\tau)$ is a discrete space. Also, for any $T_1$ topological space $(X, \tau)$, $dt(\overline{G(\tau)})=2$ if and only if $X$ has at least one isolated point. Finally, if $(X, \tau_X)$ and $(Y, \tau_Y)$ are discrete topological spaces such that $|X|\geq 3$ and $|Y|\geq 3$, then $\overline{G(\tau_X)}$ is isomorphic to $\overline{G(\tau_Y)}$ if and only if $X$ and $Y$ are homeomorphic if and only if $L(\overline{G(\tau_X)})$ is isomorphic to $L(\overline{G(\tau_Y)})$. https://comb-opt.azaruniv.ac.ir/article_15028.html A weak signed total Italian dominating function (WSTIDF) of a digraph $D$ with vertex set $V(D)$ is defined as afunction $f:V(D)\rightarrow\{-1,1,2\}$ having the property that $\sum_{x\in N^-(v)}f(x)\ge 1$ for each $v\in V(D)$, where $N^-(v)$ consists of all vertices of $D$ from which arcs go into $v$. The weight of a WSTIDF is the sum of its function values over all vertices. The  weak signed total Italian domination number of $D$, denoted by $\gamma_{wstI}(D)$, is the minimum weight of a WSTIDF on $D$. We initiate the study of the weak signed total Italian domination number in digraphs, and we  present different sharp bounds on $\gamma_{wstI}(D)$. In addition, we determine the weak signed total Italian domination number of some classes of digraphs. https://comb-opt.azaruniv.ac.ir/article_15031.html In this paper, we introduce and investigate the join standard graph $G_S(L)$ of a finite lattice $L$. We explore structural properties of the graph such as connectedness, girth, and provide necessary and sufficient conditions for the existence of universal and isolated vertices. We show that a lattice homomorphism $\varphi$ from $L_1$ to $L_2$ induces a graph homomorphism between $G_S(L_1)$ and $G_S(L_2)$. We further analyze the relationship between the graph of a lattice product and the product of graphs of its constituent lattices. Subsequently, we establish a condition under which the graph becomes hypertriangulated. We prove that the graph $G_S(L)$ is complemented if and only if the underlying lattice has cardinality at most two. Finally, we provide a criterion under which the subgraph $G_S(L)-1$ becomes disconnected. https://comb-opt.azaruniv.ac.ir/article_15037.html For a connected graph $G$ with $n>2\Delta$ vertices, $m$ edges, and maximum degree at most $\Delta\geq 3$, we show $i(G)\leq \left(1-\Omega\left(\frac{1}{\Delta^4}\right)\right)n-\frac{m}{\Delta}+O\left(\frac{1}{\Delta^2}\right)$ and discuss related problems.   https://comb-opt.azaruniv.ac.ir/article_15038.html Let $D$ be a digraph with order $n$ and $a$ arcs. Let $d_1^{+}, d_2^{+},\dots, d_n^{+}$ be the vertex outdegrees of $D$. The first outdegree Zagreb index of $D$ is denoted by $Zg^{+}(D)$ and is defined as $Zg^{+}D)=\sum\limits_{i=1}^{n}(d_i^{+})^2$. In this paper, we completely characterize the oriented graphs which attain the maximum value for the first outdegree Zagreb index $Zg^{+}(D)$ among all connected oriented graphs $D$ of order $n$ with $n-1\le a\le 2n-3$. Further, we determine the oriented graphs which attain the second maximum value for $Zg^{+}(D)$ among all oriented graphs of order $n$ with $n-1\le a\le n+2$. We consider the problem of determining the orientations which attain the maximum and the minimum values for the first outdegree Zagreb index for the Path, the Cycle and the Star. https://comb-opt.azaruniv.ac.ir/article_15039.html Vertex energy is a local spectral invariant that measures the contribution of individual vertices to the overall energy of a graph. Understanding how vertex energy behaves under graph transformations is essential for both theoretical insights and practical applications in spectral graph theory and network analysis. In this paper, we investigate the preservation of vertex energy under two fundamental graph constructions: the double graph and the bipartite double cover. We prove that for any connected graph \( G \), the vertex energies of the duplicated vertices in both \( \mathrm{D}(G) \) and \( \mathrm{DC}(G) \) remain identical to those in \( G \). These results demonstrate the robustness of vertex energy as a spectral measure invariant under these duplication operations. To illustrate the theorems, we provide explicit examples and computational verifications using SAGEMATH, with code publicly available for reproducibility. Our findings contribute to the deeper understanding of spectral properties in graph operations and open avenues for further research in spectral invariants of graph transformations. https://comb-opt.azaruniv.ac.ir/article_15040.html A dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A total dominating set in $G$ is a dominating set $S$ with the additional property that the subgraph $G[S]$ induced by $S$ is isolate-free. A triad dominating set $S$ (also called a $3$-component dominating set in the literature) is a dominating set in which every component in $G[S]$ has order at least~$3$. The triad domination number, denoted $\gamma_{td}(G)$, of $G$ is the minimum cardinality among all triad dominating sets of $G$. We observe that $\gamma(G) \le \gamma_t(G) \le \gamma_{td}(G)$, where $\gamma(G)$ is the domination number of $G$ and $\gamma_t(G)$ is the total domination number of $G$. We show that the ratio $\frac{\gamma_{td}(G)}{\gamma_t(G)}$ is at most $\frac{3}{2}$. We establish properties of the graphs $G$ satisfying $\gamma_{td}(G) = \frac{3}{2}\gamma_t(G)$ and characterize the trees achieving this equality. https://comb-opt.azaruniv.ac.ir/article_15041.html The concept of domination stability in graphs was introduced in 1983 by Bauer, Harary, Nieminen and Suffel and has been further studied by Nader Jafari Rad, Elahe Sharifi, Marcin Krzywkowski. The $\gamma^+$-stability of $G$, denoted by $\gamma^+(G)$, is the minimum number of vertices whose removal from $G$ increases the domination number. The $\gamma^-$-{\it stability} of $G$, denoted by $\gamma^-(G)$, is the minimum number of vertices whose removal from $G$ decreases the domination number. The {\it domination stability} of $G$, denoted by $st_\gamma(G)$, is the minimum number of vertices whose removal changes the domination number. In this paper the concept of domination stability is extended to $m$-eternal domination. {\it Eternal domination} of a graph requires the vertices of a graph to be protected, against infinitely long sequences of attacks, by guards located at vertices (at most one guard in each vertex), and a guard must move from a neighboring vertex to an attacked vertex with the requirement that the configuration of guards induces a dominating set at all times. Two models of the problem, one in which only one guard moves at a time and one in which more than one guard may move simultaneously are studied in the literature. The model of eternal domination in which more than one guard move simultaneously is called the $m$-eternal domination. The $m$-eternal domination number, $\gamma_m^\infty(G)$ of a graph $G$ is the minimum number of guards needed to defend $G$ against any such sequence of attacks. https://comb-opt.azaruniv.ac.ir/article_15042.html The Sombor index of a graph $G$ is a degree-based graph structure descriptor, defined as $SO(G)=\sum_{uv \in E(G)}\sqrt{d(u)^2+d(v)^2},$ in which $d(x)$ is the degree of the vertex $x \in V(G)$, for $x=u, v$. In this paper, we find a sharp lower bound of the Sombor index in trees with fixed total domination number and we characterize the extremal trees. More precisely, given any tree $T$ with order $n$ and total domination number $\gamma_t$, we prove that$SO(T)\geq \left(2\sqrt{13}+\sqrt{5}-\frac{7\sqrt{2}}{2}\right)(n-2\gamma_t)+4\sqrt{2}\gamma_t+2\sqrt{5}-6\sqrt{2}.$This lower bound improves, in many cases, the known lower bounds given with the order and with the order and the domination number of the tree. https://comb-opt.azaruniv.ac.ir/article_15044.html The intersection graph of an algebraic structure plays a pivotal role in understanding and analyzing algebraic structures---such as groups, rings, modules, acts---by encoding substructural relationships into graph-theoretic frameworks. In this paper, we introduce a new intersection-graph type for an $S$-act $A$ over a semigroup $S$, termed the \emph{simple intersection graph} of $A$, denoted by $GS(A)$. We focus on the relationship between algebraic properties of $A$ and graph-theoretic characteristics of $GS(A)$, including degree, cycles, cliques, connectivity, bipartiteness and dominaning sets. Specifically, we characterize $S$-acts $A$ for which $GS(A)$ is complete, connected or complete bipartite, and determine key invariants such as degree, girth, diameter, clique number and domination number of $GS(A)$. Applications include solutions to coloring optimization problems and extensions to semigroup-based graphs $GS(S)$. https://comb-opt.azaruniv.ac.ir/article_15045.html In this paper, an algorithmic approach is explored towards vertex coloring, coprime labeling, and coprime index of certain variant of the dot product graphs. The notion of coprime index $\mu(G)$ of a graph $G$ was introduced by Katre et al. This notion has interesting connections with the clique number $\omega(G)$, the intersection number $i(G)$ of $G$, and the edge-clique covering number $\theta_e(G^\complement)$ of the complement of the graph. Katre et al. posed a problem to characterize the graphs for which clique number and coprime index are equal, i.e., $\omega(G)=\mu(G)$.In this paper, we provide a broader class of combinatorial graphs $G(R_n)$ satisfying this equality. With a slight modification, these graphs are the dot product graphs introduced by Badawi. The graph $G(R_n)$ is associated to a subset $R_n$ of the first octant of $\mathbb R^n$, instead of associating to a ring. This graph generalizes the Kneser graphs, the Boolean graphs, and more generally, the zero-divisor graph of the ring $\mathbb F_{q_1} \times \mathbb F_{q_2} \times \dots \times \mathbb F_{q_n}$. We first explore the structure of the graph $G(R_n)$ recursively using $G(R_{n-1})$. Then, we utilize it to obtain simple algorithms for the graph labelings such as vertex coloring and coprime labeling of these graphs, and show that these labelings are minimal. The chromatic number $\chi(G(R_n))$ and the coprime index $\mu(G(R_n))$ of $G(R_n)$ are determined. Consequently, we have the class of graphs $G(R_n)$ satisfying the equality: $ \omega(G(R_n))= \mu(G(R_n))=\chi (G(R_n))=\theta_e(G(R_n)^\complement)=i(G(R_n)^\complement)$. https://comb-opt.azaruniv.ac.ir/article_15046.html A 1-factorization is a partition of the edge set of a graph into perfect matchings. The concept of 1-factorization is of great interest due to its applications in modeling sports tournaments. An invariant of a 1-factorization is a property that depends only on its structure such that isomorphic 1-factorizations are guaranteed to have equal invariant values. As such, non-isomorphic 1-factorizations may or may not have different invariant values. An invariant is complete when any two non-isomorphic 1-factorizations have distinct invariant values. We review seven invariants available in the literature to distinguish non-isomorphic 1-factorizations of $K_{2n}$ (complete graphs with an even number of vertices). Additionally, considering that the invariants available in the literature are not complete, we propose two new ones, denoted lantern profiles and even-size bichromatic chains. We analyze the invariants concerning their sizes and calculation time complexity. Furthermore, we conduct computational experiments to assess their ability to distinguish non-isomorphic 1-factorizations. To accomplish that we use the sets of non-isomorphic 1-factorizations of $K_{10}$ and $K_{12}$. We also consider the sets of non-isomorphic perfect 1-factorizations of $K_{12}$, $K_{14}$, and $K_{16}$, as well as randomly generated 1-factorizations of $K_{16}$ and $K_{20}$. Moreover, the experiments evaluate how the combination of invariants can increase the distinguishing ability.   https://comb-opt.azaruniv.ac.ir/article_15048.html Constacyclic codes constitute a significant class of linear codes in coding theory and play a crucial role in the construction of optimal codes. Several optimal linear codes have been derived from constacyclic codes. In 2015, Ashraf and Mohammad investigated $(1+2u)$-constacyclic codes over $\mathbb{Z}_{4}+u\mathbb{Z}_{4}$ with $u^{2}=0$. More recently, G. Karthick studied $(1+2u+2v)$-constacyclic codes over the semi-local ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ under the conditions $u^{2}=3u$, $v^{2}=3v$, and $uv=vu=0$. In this paper, we generalize their results by examining $(1+2u_{2}+2u_{3}+\dots+2u_{t})$-constacyclic codes over the semi-local ring $\mathcal{S} = \mathbb{Z}_{4} + u_{2} \mathbb{Z}_{4} + u_{3} \mathbb{Z}_{4} + \dots + u_{t} \mathbb{Z}_{4}$, where $u_{i}^{2} = k u_{i}$ and $u_{i} u_{j} = u_{j} u_{i} = 0$ for $2 \leq i \leq t$, $i \neq j$, with $u_{1}=1$ and $k \in \mathbb{Z}_{4}$. We focus on $(1+2u_{2}+2u_{3}+\dots+2u_{t})$-constacyclic codes over $\mathcal{S}$ and establish their structural properties. By introducing new Gray maps, we demonstrate that these constacyclic codes can be transformed into cyclic and quasi-cyclic codes over $\mathbb{Z}_{4}$. Furthermore, we characterize a generating set for these codes when the code length is odd. %nand provide explicit examples illustrating their construction. Our findings contribute to the database of $mathbb{Z}_{4}$ codes and enhance the understanding of constacyclic codes over extended non-chain rings. https://comb-opt.azaruniv.ac.ir/article_15050.html The neighborhood inverse sum indeg index, denoted as $ISI_N(\mathcal{G})$, of a simple graph $\mathcal{G}$ is defined as the sum of the terms $( \frac{\mathscr{\delta}_{\mathcal{G}}(\mathscr{r})\mathscr{\delta}_{\mathcal{G}}(\mathscr{s})}{\mathscr{\delta}_{\mathcal{G}}(\mathscr{r})+\mathscr{\delta}_{\mathcal{G}}(\mathscr{s})} \) for all edges \( \mathscr{\mathscr{rs}} )$ in $\mathcal{G}$. Here, \( \mathscr{\delta}_{\mathcal{G}}(\mathscr{r}) \) represents the neighborhood degree of a vertex \( \mathscr{r} \), which is the sum of the degrees of the neighbours of \( \mathscr{r} \) in \( \mathcal{G} \). This article establishes bounds for the \( ISI_N(\mathcal{G}) \) index in relation to various graph invariants and its connection to neighborhood degree-sum-based topological indices. We also present results on the \( ISI_N \) index concerning different graph operations. Furthermore, we analyze the physico-chemical properties of 55 benzenoid hydrocarbons and validate our model with 10 more benzenoid hydrocarbons. https://comb-opt.azaruniv.ac.ir/article_15051.html A word w is a finite sequence of symbols belonging to a finite set, called an alphabet. A scattered subword of a word w is a subsequence of w. The Parikh matrix of a word w over an ordered alphabet with an ordering on its elements, is an upper triangular matrix with its entries giving the counts of different occurrences of certain scattered subwords in the word w. Based on the notions of scattered subword and Parikh matrix, several properties of images of words under morphisms have been established. Here we consider Dejean morphism on three letters and derive several properties for images of binary words under this morphism in the context of Parikh matrices. https://comb-opt.azaruniv.ac.ir/article_15054.html In this paper‎, ‎we consider a group of cactus graphs with specific pendant edge structures such as sparkle‎, ‎sun‎, ‎and broken sun graphs‎. ‎We focus on calculating the extremal Sombor index of these graphs due to their importance in various scientific fields‎. ‎The Sombor index‎, ‎a topological index predicting physicochemical properties of molecules‎, ‎was introduced by Ivan Gutman‎. ‎This study extends previous work on Sombor indices for cactus graphs‎, ‎presenting new findings on the Sombor index of these specialized graphs and their potential use in molecular property prediction‎. https://comb-opt.azaruniv.ac.ir/article_15060.html Let $G=(V, E)$ be a graph of order $n$. Let $D\subseteq\{0,1,2,\dots, \text{diam}(G)\}$ be nonempty. The $D$-neighborhood $N_D(x)$, of a vertex $x$ is the set of all vertices whose distance from vertex $x$ is an element in $D$, that is, $N_D(x)=\{y\in V:\ d(x,y)=m, m\in D\}$. A $D$-distance magic labeling of $G$ is a bijection $f\colon V\to \{1,2,\dots,n\}$ for which there exists a positive integer $k$, such that $\sum_{x\in N_D(v)}f(x)=k$ for all $v\in V$, where $N_D(v)$ is the $D$-open neighborhood of $v$. Let $\Gamma$ be an abelian group of order $n$. A $(\Gamma,D)$-distance magic labeling of $G$ is a bijection $l\colon V\to \Gamma$ for which there exists an element $\mu\in \Gamma$, such that $\sum_{x\in N_D(v)}l(x)=\mu$ for all $v\in V$. This paper presents the necessary and sufficient conditions for the existence of $D$-distance magic labeling for $C_n^r$ for a set $D$ containing elements in arithmetic progression. For the same set $D$, we also study the $(\Gamma, D)$-distance magic labeling of $C_n^r$ for some specific classes of abelian groups $\Gamma$. https://comb-opt.azaruniv.ac.ir/article_15061.html The use of local metric resolvability can be realized in delivery services for optimal placement of existing and new resources like medical facilities, stores, and fire stations. The local metric basis produces codes for the facilities and regions to be served by these facilities in a network or a graph in such a way that the adjacent nodes get unique codes in terms of distances, so that each facility is used optimally. In this paper, the local metric dimension (LMD) has been computed for convex polytopes $B_n$, $C_n$, $D_n$, and $Q_n$. An algorithm to extend the number of resources in a distributed network and real-life applications of local metric resolvability have also been investigated. https://comb-opt.azaruniv.ac.ir/article_15062.html A set of vertices $S\subseteq V$ in a graph $G=(V,E)$ is called an internal majority set if for every vertex $v\in S$, a majority of the neighbors of $v$ are in $S$, or equivalently, every vertex $v\in S$ has fewer neighbors in $V-S=\overline{S}$ than it has in $S$. A set $S$ is called an external majority set if for every vertex $v\in\overline{S}$, a majority of the neighbors of $v$ are in $S$, or equivalently, every vertex $v\in\overline{S}$ has more neighbors in $S$ than it has in $\overline{S}$. A set of vertices $S\subseteq V$ in a graph $G=(V,E)$ is called a total majority set if for every vertex $v\in V$, a majority of the neighbors of $v$ are in $S$, or equivalently, every vertex $v\in V$ has more neighbors in $S$ than it has in $\overline{S}$. In this paper we show that majority sets in graphs are closely related to, but different than, a variety of sets that have been studied, such as offensive alliances, cost effective and very cost effective sets and unfriendly partitions in graphs. We also prove that the decision problems associated with external majority sets and total majority sets are NP-complete. Finally, we present a list of open problems related to majority sets in graphs. https://comb-opt.azaruniv.ac.ir/article_15063.html Cylindrical graphs and torus grid graphs are naturally constructed from sub-graphs of the infinite grid by certain identifications of boundary vertices. Considering various domination type problems, it is usually possible to find an optimal solution on the infinite grid. To the contrary, exact values of invariants for the cylindrical and torus grid graphs are typically only known for special subfamilies, and are in general hard to compute. The 4-domination and 4-rainbow domination of cylindrical graphs is studied, and some new formulae and improved bounds are reported, generalizing recent results for the case $k = 2$ in [Computational and Applied Mathematics 44(5), 293 (2025)]. We also consider weak 4-domination and singleton 4-rainbow domination. https://comb-opt.azaruniv.ac.ir/article_15073.html This paper focuses on three types of product graphs, analyzing the relationships between their ECD sets, SDCTD sets, and factor graphs, while also constructing such ECD sets and SDCTD sets for the product graphs in question. We establish the necessary and sufficient conditions for an ECD set of a hypercube to be an SDCTD set, clarify the conditions under which an SDCTD set constitutes a PD set in a general graph $G$, and further deduce these conditions specifically for regular graphs and bipartite graphs. By hierarchically partitioning SDCTD sets via tree decomposition, we derive new upper bounds for the domination number of the modular product of two graphs, as well as for that of the modular product of two regular graphs. Additionally, we fully characterize all graph pairs $(G,H)$ for which $\gamma(G\diamond H)=5$ and prove a general lower bound  $\gamma(G\diamond G)\geq\gamma(G)+2$. The paper concludes by outlining several avenues for potential future research. https://comb-opt.azaruniv.ac.ir/article_15074.html The atom-bond sum-connectivity index ($\operatorname{ABS}$) has recently been introduced as a variation of the classical atom-bond connectivity index, in which the product of vertex degrees in the denominator of each term is replaced by their sum. In this paper, we first establish a connection between the $\operatorname{ABS}$ index and the spectral radius of a graph. We then examine the relationships between the $\operatorname{ABS}$ index and several well-known degree-based topological indices. Furthermore, we improve several previous bounds presented in the literature. https://comb-opt.azaruniv.ac.ir/article_15075.html Let $H$ be a graph with vertex set $V.$ An Italian dominating function (IDF) on $H$ is a function from $V$ to the set $\{0,1,2\}$ having the property that any vertex assigned $0$ is adjacent to two vertices assigned $1$ or one vertex assigned $2.$ The value $\sum_{x\in V}h(x)$ is called the weight of an IDF $h$ on $H.$ A global Italian dominating function (GIDF) on $H$ is an IDF on $H$ and its complement. The minimum weight of an IDF (resp., GIDF) on $H$ is the Italian (resp., global Italian) domination number of $H.$ In this paper, we establish several relations between the global Italian domination and Italian domination numbers. In particular, we determine the difference between these two parameters of cubic graphs. https://comb-opt.azaruniv.ac.ir/article_15085.html The MacWilliams identity provides a fundamental link between the weight enumerator of a linear code and its dual. While generalizations exist for joint weight enumerators and $\lambda$-ply weight enumerators, a unifying framework encompassing these extensions has remained elusive. In this paper, we introduce the $\lambda$-ply joint weight enumerators and obtain a novel generalization of the MacWilliams identity that subsumes previously known results as special cases. https://comb-opt.azaruniv.ac.ir/article_15088.html We say a proper coloring  of a graph is distance-$k$ fall if every vertex is within distance $k$ of at least one vertex of every color. We show that if $G$ is a connected graph of order at least $3$ that is $3$-colorable, then it has a distance-2 fall 3-coloring. Further, for every integer $k\ge 2$, if $T$ is a tree of order at least $k$, then $T$ has a $k$-coloring such that every vertex is within distance $k-1$ of every color. This proves an old conjecture of Beineke and Henning that every tree of order $n$ has an independent distance-$d$-dominating set of size at most $n/(d + 1)$. https://comb-opt.azaruniv.ac.ir/article_15089.html The $KG$-Sombor index of a graph $G$ is defined as $$KG(G)=\sum_{\substack{u \in V(G) \\ e \in E(G) \\ u \sim e}}\sqrt{d_{G}^{2}(u)+d_{G}^{2}(e)},$$ where the summation goes over pairs of vertices $u$ and edges $e$ such that $e$ is incident to $u$. In this paper, we establish sharp upper bounds for the $KG$-Sombor index in three classes of graphs: graphs of order $n$ with $k$ pendent vertices, graphs of order $n$ with $k$ cut edges, and unicyclic graphs of order $n$ with girth $g$. Moreover, in each case, the extremal graphs attaining these bounds are completely characterized. https://comb-opt.azaruniv.ac.ir/article_15090.html In this paper, we give some properties of the generalized commutative Leonardo quaternions, among others the Binet formula, generating function, and the general bilinear index-reduction formula which imply d'Ocagne, Vajda, Halton, Catalan, and Cassini identities. We also give the matrix representations and some sum formulas of the generalized commutative Leonardo quaternions. Moreover, we present a one-parameter generalization of the generalized commutative Leonardo quaternions and their properties.  https://comb-opt.azaruniv.ac.ir/article_15094.html The Estrada index is a spectral invariant with wide applications in graph theory and network science, and has been studied for both unsigned and signed graphs. In this paper, we investigate its extremal behavior over all signatures of a fixedunderlying graph. We prove that for any connected graph balanced signatures are the only ones that maximize the Estrada index. https://comb-opt.azaruniv.ac.ir/article_15095.html A subset of vertices $S$ of a graph $G$ is a dominating set if every vertex in $V \setminus S$ has at least one neighbor in $S$. A domatic partition is a partition of the vertices of a graph $G$ into disjoint dominating sets. The domatic number $d(G)$ is the maximum size of a domatic partition. We consider the number of domatic partitions of $G$ with different sizes. Inspired by existing results for trees, this paper extends the analysis to several other important families of graphs. We focus primarily on the coefficient $dp(G,2)$, which counts domatic 2-partitions. We present some recurrence relations for this coefficient for the cycle graphs $C_n$ and the wheel graphs $W_n$. Furthermore, we present precise closed-form formulas for the domatic polynomial of star graphs $K_{1,n}$ and friendship graphs $F_n$. We also derive a formula for $dp(K_{m,n}, 2)$ for complete bipartite graphs. Finally, through a comprehensive computational analysis of all $3$-regular graphs of order $10$, we observe that the Petersen graph cannot be determined by its domatic polynomial. https://comb-opt.azaruniv.ac.ir/article_15098.html In this paper, we investigate the second inverse sum indeg index $ISI_2$ of graphs, a topological index that has significant applications in chemical graph theory. Upper and lower bounds for $ISI_2$ of graphs and trees with a specified number of pendent edges are established. Furthermore, $ISI_2$ of various bridge graphs are computed. The main contribution of this work lies in presenting precise bounds and exact expressions for particular families of graphs, offering resources for researchers and engineers in mathematical chemistry and applied graph theory. https://comb-opt.azaruniv.ac.ir/article_15101.html This study focuses on the low-complexity implementation of semidefinite relaxation (SDR) to generate bounds for the Boolean Quadratic Programming Problem with Generalized Upper Bound Constraints (BQP-GUB). Most current SDR approaches rely on interior-point methods (IPM), which, despite having worst-case polynomial complexity, can be computationally expensive in practice. We depart from the IPM framework and investigate the use of other low per-iteration-complexity techniques for the solution of BQP-GUB. Specifically, we apply the row-by-row (RBR) method, called NuclearRBR, to solve the semidefinite programs that emerge from reformulating the BQP-GUB as an unconstrained Boolean Quadratic Programming Problem (UBQP). In this formulation, a nonconvex rank-one constraint is relaxed by a convex nuclear norm constraint. The RBR method only requires matrix-vector multiplications in each iteration, making it highly efficient. Numerical results demonstrate that NuclearRBR outperforms the semidefinite dual (SDD) method and other similar existing methods like SDcutRBR method [R.K. Nayak and N.K. Mohanty, Improved row-by-row method for binary quadratic optimization problems, Ann. Oper. Res. 275 (2019), 2, 587–605]. https://comb-opt.azaruniv.ac.ir/article_15113.html A graph $G$ is a \emph{cover} of a graph $F$ if there exists an onto mapping $\pi : V(G) \to V(F)$, called a (\emph{covering}) \emph{projection}, such that $\pi$ maps the neighbours of any vertex $v$ in $G$ bijectively onto the neighbours of $\pi(v)$ in $F$. This paper is the first attempt to study the connection between domination parameters and graph covers. We focus on the domination number, the total domination number, and the connected domination number. We prove upper and lower bounds for the domination parameters of $G$. Moreover, we propose a conjecture on the lower bound for the domination number of $G$ and provide evidence to support the conjecture. https://comb-opt.azaruniv.ac.ir/article_15115.html Among the defined $148$ discrete Adriatic indices, the max-min rodeg index is one. It is a good predictor for the enthalpy of vaporization and standard enthalpy of vaporization for octane isomers, as well as the log water activity coefficient for polychlorobiphenyls.  For a graph $G$, here we concentrate on the max-min rodeg index, defined as \begin{equation*}Mm_{sde}(G)=\sum_{x\sim y}\sqrt{\frac{max\{d_x, d_y\}}{min\{d_x, d_y\}}},\end{equation*}where $x\sim y$ and $d_x$ represents the adjacency of two vertices $x$ and $y$, and the degree of the vertex $x$, respectively. First, we present some bounds for the max-min rodeg index via standard inequalities. Then we provide upper bounds via some graph parameters for the max-min rodeg index of $G$. Also, we obtain a relation between the max-min degree index and the energy of $G$. Finally, we study the extremal value problem over chemical graphs concerning the max-min rodeg index. https://comb-opt.azaruniv.ac.ir/article_15116.html A graph $G$ is $k$-stepwise irregular if $|d_G(u)-d_G(v)|= k$ holds for every edge $uv$ of $G$. It is proved that for such a graph $m(G) \leq (n(G)^2 - k^2)/4$ holds, where the equality holds if and only if $G\cong K_{\frac{n(G)+k}{2},\frac{n(G)-k}{2}}$. Using this result, sharp lower and upper bounds are derived for Zagreb (co)indices, the Sombor index, and the Randi\'c index of $k$-stepwise irregular graphs. https://comb-opt.azaruniv.ac.ir/article_15117.html We consider the exprissibility in monadic second order logic of certain relations of importance in computer science. For integers $n\geq 1$ and $k\leq b$, a $k$-tuple of sequences in $\{0,1,\ldots, b-1\}^n$ are said to be $k$-hashed if there is a coordinate where they all differ. A set $\mathcal{C}$ of sequences is said to be a $k$-hash code if any $k$ distinct elements are $k$-hashed. Testing whether a code is $k$-hashing and determining the largest size of $k$-hash codes is an important problem in computer science. The use of general purpose solvers for this problem leads to question what minimal logic is needed to represent the problem.In this paper, we prove that the $k$-hashing relation on $k$-tuples is not definable in Monadic Second Order Logic (MSO), highlighting its limitations for this problem. Instead, the property can be expressed in extensions of the MSO that add the equi-cardinality relation.Finally, since to the best of our knowledge there are no available solvers for MSO with such global cardinality constraints,we provide a practical SMT encoding in Z3 for fixed parameters. In addition, we computationally recover the non-existence of a trifferent code of size $11$ and length $5$. https://comb-opt.azaruniv.ac.ir/article_15121.html Let $G = (V, E)$ be a simple undirected connected graph. A set $C \subseteq V(G)$ is weakly convex in $G$ if for every two vertices $u,v$ in $G$, there exists a $u-v$ geodesic whose vertices are in $C$. A set $C \subseteq V$ is an outer-weakly convex dominating set if every vertex not in $C$ is adjacent to some vertex in $C$ and the set $V(G)\setminus C$ is weakly convex in $G$. The outer-weakly convex domination number of graph $G$, denoted by $\widetilde{ \gamma}_{wcon}(G)$, is the minimum cardinality of an outer-weakly convex dominating set of graph $G$. In this paper, we determine the outer-weakly convex domination number of two graphs under the Cartesian, strong and lexicographic products, and discuss some important combinatorial findings. https://comb-opt.azaruniv.ac.ir/article_15122.html The Roman domination problem is a combinatorial optimization problem on a graph asking to assign a label from $\{0,1,2\}$ to each vertex feasibly, such that the total sum of assigned labels is minimized. Let $G=(V,E)$ be a graph, and let \(U_1, U_2\subseteq V\) be two non-empty disjoint subsets. We say that the pair \( \{U_1,U_2\}\) is Roman-feasible if there exists a Roman dominating function \(f:V\to \{0,1,2\}\) such that \(V_0=V\setminus (U_1\cup U_2),\) \(V_1=U_i\) and \(V_2=U_{3-i}\) for some \(i\in \{1,2\},\) where $V_j$ denotes the set of vertices assigned label $j$ by $f$. The set \(\{U_1,U_2,U_3\}\) is a Roman coalition if the following two conditions hold: (i) the pair \(\{U_i,U_j\}\) is not Roman-feasible for any different \(1\le i,j\le 3;\) (ii) there exists \(k\) such that the pair \(\{U_i\cup U_j, U_k\}\) is Roman-feasible, where \(\{i,j,k\}=\{1,2,3\}.\) The purpose of this paper is to introduce and study the new concept of Roman coalitions in graphs, providing basic properties, lower and upper bounds, as well as exact values in specific cases. https://comb-opt.azaruniv.ac.ir/article_15123.html Covering arrays (CAs) are widely recognized combinatorial designs that facilitate efficient test suite generation in software testing. For a positive integer $n$ and a set $S$ of size $g$, a \emph{covering array on a hypergraph} $H = (V, E)$ of size $n$ and alphabet size $g$, denoted $CA(n, H, g)$, is an $n \times |V|$ array with entries from $S$, where columns correspond to the vertices in $V$, such that for every hyperedge $e \in E$ of size $m$, the sub-array indexed by the vertices of $e$ contains all the ordered $m$-tuples from $S^m$ at least once, as a row. The smallest $n$ for which such an array exists is the \emph{covering array number of $H$}, denoted $CAN(H, g)$. This parameter represents the minimal test suite size for practical applications. For a $t$-uniform hypergraph $H$ and positive integers $g$ and $n$, a $CA(n, H, g)$ exists if and only if there is a hypergraph homomorphism from $H$ to $t\text{-}QI(n, g)$, a distinctive class of hypergraphs called \emph{qualitative independence hypergraphs}. Consequently, for such a hypergraph $H$, $CAN(H, g) \leq CAN(t\text{-}QI(n,g),g)$. In this paper, we develop a technique to determine the strong chromatic number and the size of the largest $3$-cliques of $3\text{-}QI(n, 2)$ to establish that $CAN(3\text{-}QI(n, 2),2)=n$ for some values of $n$ and apply the results obtained to find the covering array number of certain classes of hypergraphs. https://comb-opt.azaruniv.ac.ir/article_15126.html This work presents a predictor-corrector interior-point algorithm for solving the weighted linear complementarity problem. By applying Newton's type method to the central path system, the search directions are obtained. The algorithm works in the $\tau$-neighborhood, which measures the proximity of iterates to the central path. By suitable choice of parameters, the global convergence of the method under mild conditions is guaranteed. The iteration bound derived  to find   $\varepsilon$-approximate  solution  matches the best known iteration bound for these types of problems. To the best of our knowledge, this is the first work based on these types of search directions. https://comb-opt.azaruniv.ac.ir/article_15128.html A set $S$ of vertices is a $[1,2]$-set of a graph $G$ if every vertex $v$ not in $S$ is adjacent to at least one but no more than two vertices in $S$. The minimum cardinality of a $[1,2]$-set is the $[1,2]$-domination number. In this paper, we present two upper bounds on the $[1,2]$-domination number of trees in terms of the order, number of support vertices and number of leaves. Furthermore, extremal trees reaching one of these two bounds are provided. https://comb-opt.azaruniv.ac.ir/article_15132.html A classic result in domination theory is that a regular graph has independent domination number at most half the order. We strengthen this result to ``almost-regular'' graphs by showing that if a graph has minimum degree $\delta > 0$ and maximum degree at most $\delta + 3$, and the subgraph induced by the vertices of degree $\delta + 3$ (if any) is bipartite, then the independent domination number is at most half the order. We also discuss related questions. https://comb-opt.azaruniv.ac.ir/article_15133.html Let $dim(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le dim(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars attain the bound, and among connected unicyclic graphs such graphs are $t$-cycles for $t\in \{3,4,5\}$. It is shown that for any $1\leq n< m$, there exists a graph $G$ with $D(G)=n$ and ${\rm dim}(G)=m$. Using the bound $D(G) \le dim(G)+1$, graphs with $D(G) = n(G)-2$ are classified.  https://comb-opt.azaruniv.ac.ir/article_15135.html Let $R$ be a commutative ring, the associate graph, $Ass(R)$ of ring $R$ has the elements of ring $R$ as vertices and two distinct vertices $u$ and $v$ are adjacent if $u$ and $v$ are associate elements of $R$. In this article we investigate the Wiener index of $Ass(\mathbb{Z}_{n})$ and its line graph for all $n\in \mathbb{N}$. We also give some characterization results regarding degree, diameter, girth, clique number, chromatic number, domination number, and independence number of $Ass(\mathbb{Z}_{n})$ and $L\left( Ass(\mathbb{Z}_{n}) \right)$. https://comb-opt.azaruniv.ac.ir/article_15137.html Let $G = (V(G), E(G))$ be a graph. A set $I \subseteq V(G)$ is independent if no two vertices of $I$ are adjacent. A set $D \subseteq V(G)$ is dominating if every vertex $u \in V(G) \setminus D$ is adjacent to a vertex in $D$. A set $L \subseteq V(G)$ is an independent locating-dominating set (ILD-set) of $G$ if $L$ is independent and dominating with the additional property that $N (u) \cap L \neq N (v) \cap L$ for any pair of distinct $u, v \in V(G) \setminus L$. The independent location-domination number of a graph $G$ is the minimum cardinality of an ILD-set of $G$ and is denoted by $i_{\ell}(G)$. In this paper, we study the non-existence of ILD-sets of maximal outerplanar graphs and circulants graphs. In trees, we prove that \textcolor{red}{$\frac{n + 1}{3} \leq i_{\ell}(T) \leq n - 1$} for every tree $T$ of $n$ vertices. We further prove that there exists a tree $T$ with prescribed value $i_{\ell}(T)$ between these bounds. https://comb-opt.azaruniv.ac.ir/article_15141.html In this paper, we determined the parameters of the unit $\mathbb Z_r$-Simplex code under the Homogeneous weight metric and showed that it is an $\left[ \frac{\rho^{k-1}}{\rho-1}, ~k~, \rho^{k-1}((p-1)p^{m-2})\right]$ $Z_r$-Simplex code if $m=2$, $\left[ \frac{\rho^{k-1}}{\rho-1}, ~k~, \rho^{k-2}((p-2)p^{2m-2}+2p^{m-1})\right]$ if $m>2$, where $r=p^m$ with rank $k$. Further, we obtain the weight distribution of the $\mathbb Z_r$-Simplex code under the Homogeneous weight metric for the particular rank $k=2$. https://comb-opt.azaruniv.ac.ir/article_15145.html Let $G$ be a graph and $S(G)$ be the Seidel matrix of $G$. Let $s_1\ge s_2\ge \dots\ge s_n$ be the eigenvalues of $S(G)$. The spread of matrix $S(G)$ defined as  $s(G) := max_{i,j}|s_i-s_j| = s_1-s_n$. The Seidel energy of $G$, denoted by $SE(G)$, is defined to be the sum of the absolute value of all eigenvalues of the Seidel matrix of $G$. Willem Haemers conjectured that the Seidel energy of any graph with $n$ vertices is at least $2n-2$. Motivated by this conjecture, we prove that the conjecture is true if $s(G)\le n$. Moreover, we present some new bounds for the Seidel energy and also we study some properties of the Seidel eigenvalues of $G$.  Our results improve some known results. https://comb-opt.azaruniv.ac.ir/article_15147.html ‎Archdeacon et al. [J. Graph Theory 46 (2004), 207--210] proved that if $G$ is a bipartite graph with partite sets $X$ and $Y$ whose vertices in $Y$ are of minimum degree at least $3$ then there exists a set $A\subseteq X$ of size at most$\frac{|X\cup Y|}{4}$ such that every vertex in $Y$ is adjacent to a vertex in $A$. We generalize this result for all bipartite graphs with minimum degree $\delta\geq 3$ using the Brooks Theorem on the vertex coloring. https://comb-opt.azaruniv.ac.ir/article_15148.html A graph is subcubic if it is connected and its maximum vertex degree does not exceed 3. Two disjoint vertex subsets of a graph $G$ form a connected coalition in $G$ if neither of them is a connected dominating set but their union is a connected dominating set. A connected coalition partition of $G$ is a partition of its vertices $\pi(G) = \{V_1, V_2,..., V_k \}$, such that each $V_i$ is either a connected dominating set consisting of a single vertex or forms a coalition with some set of $\pi(G)$. The formation of connected coalitions is described by a coalition graph whose vertices correspond to the sets of $\pi$, and two vertices are adjacent if and only if the corresponding sets form a coalition in $G$. We characterize all coalition graphs of subcubic graphs. https://comb-opt.azaruniv.ac.ir/article_15151.html A vertex $v$ of a connected graph $G$ is said to be a boundary vertex of $G$ if for some other vertex $u$ of $G$, no neighbor of $v$ is further away from $u$  than $v$. The boundary  $\partial(G)$ of  $G$ is the set of all of its boundary vertices. The boundary distance matrix $\hat{D}_G$ of a graph $G=([n],E)$ is the square matrix of order $\kappa$, being $\kappa$ the order of $\partial(G)$, such that for every $i,j\in \partial(G)$,  $[\hat{D}_G]_{ij}=d_G(i,j)$. In a recent paper [doi.org/10.7151/dmgt.2567], it was shown that if a graph $G$ is either a block graph or a unicyclic graph, then $G$ is uniquely determined by the boundary distance matrix $\hat{D}_{G}$  of $G$, and it was also conjectured that this statement holds for every connected graph $G$, whenever both the order $n$ and the $|\partial(G)|$ (and thus also the boundary distance matrix) of $G$ are prefixed (i.e., the set of boundary vertices is known as a subset of $V$, but not their identities). After proving  that this conjecture is  true for several graph families, such as being of diameter 2, having  order at most $n=6$ or being Ptolemaic, we show that this statement does not hold when considering, for example, either  the family of split graphs of diameter 3 and order at least $n=10$ or the family of distance-hereditary graphs of order at least $n=8$.    https://comb-opt.azaruniv.ac.ir/article_15152.html Given a graph $\Gamma\left(V, E\right)$ with a non-empty vertex set $V$ and edge set $E$. A total $d$-labeling is the allocation of positive integers to the collection $V(\Gamma) \cup E(\Gamma)$. A labeling is termed distance vertex irregular total d-labeling (DVITL) if any two distinct vertices in $V$ possess different weights. The weight of the vertex $u \in V$ is the aggregate of labels of the neighbor vertices and the labels of edges that incident at the vertex $u$. The least number $d$ for which there exists a DVITL of $\Gamma$ is referred to as the total distance vertex irregularity strength of $\Gamma$ symbolized by $\operatorname{tdis}(\Gamma)$. In this research, we compute the precise values of the total distance vertex irregularity strength for prism graphs, web graphs, and web graphs without center. https://comb-opt.azaruniv.ac.ir/article_15153.html A vertex-labelling $f:V\to \{0,1,\ldots, |E|\}$  for a finite undirected simple graph $G(V, E)$ is called graceful  if $f$ is injective and satisfies the additional property that $\{|f(u)-f(v)|: \text{for every edge} uv\in E\}=\{1,2,\ldots,|E|\}$, where $|E|$ is the number of edges in $G$. Let $M$ be a maximum matching in $G$ and let $f$ also satisfy the property that $f(u)+f(v)=W$ for every $uv\in M$, where $W$ is a constant; then the labelling $f$ is called nicely graceful.  Furthermore,  if  $M$ is a perfect matching in $G$, then $f$ is said to be strongly graceful. In this paper, we investigate nicely and strongly graceful labellings of cycles and tadpoles that are obtained from a cycle by attaching a path to a vertex of the cycle. This leads to a complete characterisation of nicely graceful cycles and strongly graceful tadpoles. https://comb-opt.azaruniv.ac.ir/article_15155.html In this paper, we introduce and analyze the spectral properties of the Graovac-Ghorbani matrix $\tilde{\mathcal{A}}$. We first calculate the Graovac-Ghorbani index for specific classes of structures, including nicely distance-balanced, vertex-transitive, and edge-transitive graphs. By defining the Laplacian matrix associated with $\tilde{\mathcal{A}}$, we prove that for any connected graph of order $n \ge 3$, the rank of this Laplacian matrix is exactly $n-1$. We establish sharp upper and lower bounds for the spectral radius of $\tilde{\mathcal{A}}$ in terms of graph parameters for various families, including trees, unicycle bipartite graphs, and general bipartite graphs, characterizing the extremal cases such as complete bipartite graphs. Furthermore, the spectral properties of strongly distance-balanced (SDB) graphs are investigated. A key finding of this study is the proof that the $\tilde{\mathcal{A}}$-eigenvalues of any bipartite graph are symmetric with respect to zero. However, we demonstrate that the converse does not hold, implying that a symmetric $\tilde{\mathcal{A}}$-spectrum does not necessarily guarantee the bipartiteness of the graph. https://comb-opt.azaruniv.ac.ir/article_15156.html This paper investigates $k$-distance magic labeling, for a positive integer $k$, within the framework of sibling graphs, that is, graphs that are both self-centered and antipodal. A $k$-distance magic labeling is a vertex labeling in which the sum of the labels on all vertices at distance $k$ from any given vertex is constant throughout the graph. We establish sufficient conditions for a sibling graph to admit a $k$-distance magic labeling, covering both regular and non-regular cases. Using these conditions, we show that several well-known families of regular sibling graphs of diameter $k$ are $k$-distance magic, including cylindrical grid graphs, cyclic grid graphs, $n$-dimensional hypercubes, circulant graphs, weak Bruhat graphs, and the Möbius--Kantor graph. In addition, we construct examples of irregular sibling graphs that admit a $2$-distance magic labeling. https://comb-opt.azaruniv.ac.ir/article_15157.html The power graph $\mathscr{P}(G)$ of a group $G$ is an undirected graph with all the elements of $G$ as vertices and where any two vertices are adjacent if and only if one is the integral power of the other. So far, no spectral results had been done for the complement of power graph on any group. In this paper, we compute the adjacency, Laplacian, and signless Laplacian eigenvalues of the complement of power graphs on finite cyclic, dihedral, and quaternion groups. Also we determine all the linearly independent eigenvectors corresponding to these eigenvalues. Moreover, we see that these eigenvectors, except possibly two, are common to all the above three type of matrices. https://comb-opt.azaruniv.ac.ir/article_15158.html Let $R$ be a finite ring with identity. The idempotent graph $I(R)$ is the graph whose vertex set consists of the non-trivial idempotent elements of $R$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx = 0$. The clean graph $Cl(R)$ is a graph whose vertices are of the form $(e, u)$, where $e$ is an idempotent element and $u$ is a unit of $R$. Two distinct vertices $(e,u)$ and $(f, v)$ are adjacent if and only if $ef = fe = 0$ or $uv = vu = 1$. The graph $Cl_2(R)$ is the subgraph of $Cl(R)$ induced by the set $\{(e, u): e \text{ is a nonzero idempotent element of } R\}$. In this study, we examine the structure of clean graphs over $\mathbb{Z}_{n}$ derived from their $Cl_2$ graphs and investigate their relationship with the structure of their idempotent graphs. Furthermore, we obtain an equivalence between the isomorphism of two clean graphs and the isomorphism of their corresponding idempotent graphs over an Artinian ring. https://comb-opt.azaruniv.ac.ir/article_15161.html In this paper, we investigate the use of computational algebraic topology and simplicial homology within the framework of Sperner’s lemma to solve box- and inequality-constrained nonlinear (nonconvex) programming problems. The homology clustering method we consider provides optimal or near-optimal candidate solutions of locally convex subdomains in the search space by estimating the homology groups of a complex constructed on a hypersurface that is homeomorphic to a complex on the objective function. Each candidate point is then chosen as a starting point for a local iterative interior-point optimization technique that performs well on the corresponding locally convex subdomain. The best solution obtained from executing parallel interior-point techniques is the global optimal solution of the problem over the nonconvex domain. In addition, we compare the proposed method with the existing topographical interior-point approach and other solvers across various test problems. The computational results demonstrate that the simplicial interior-point algorithm performs well in finding the global minimum and outperforms both the topographical interior-point algorithm and the MIDACO solver in practice. https://comb-opt.azaruniv.ac.ir/article_15162.html Let $T$ be a tree of diameter at most $5$. We investigate homological invariants of its edge ideal, including the projective dimension, the Castelnuovo–Mumford regularity, and the graded Betti numbers. For trees of diameter at most $3$, all nonzero Betti numbers lie on the linear strand. For trees of diameter~$4$ and~$5$, we determine the regularity and the projective dimension explicitly. In the case of caterpillar trees of diameter~$4$, we compute all graded Betti numbers and provide an explicit formula relating them to the $f$-vector of the independence complex. Our results refine the combinatorial description of Betti numbers for forests and highlight structural features of trees with small diameter. https://comb-opt.azaruniv.ac.ir/article_15163.html The Laplacian spread of a simple graph is defined as the difference between the largest and the second-smallest eigenvalues of its Laplacian matrix. In this work, we derive bounds for the Laplacian spread, providing conditional refinements of Theorems $3.1$ and $4.1$ in [X. Chen and K.C. Das, Some results on the Laplacian spread of a graph, Linear Algebra Appl. 505 (2016), 245–260]. In addition, we present examples that illustrate the independence of the obtained bounds and show that, for these examples, each bound yields a sharper estimate than the corresponding result in [X. Chen and K.C. Das, Some results on the Laplacian spread of a graph, Linear Algebra Appl. 505 (2016), 245–260]. https://comb-opt.azaruniv.ac.ir/article_15164.html A novel invariant based on vertex degree was recently proposed by Gutman, called the Sombor index, and defined as  \begin{equation*}  \mathcal{SO}(\mathcal{G})=\sum_{v_{1}v_{2}\in \mathcal{E}(\mathcal{G})}\sqrt{\text{d}(v_{1})^2+\text{d}(v_{2})^2},\end{equation*}where $ \text{d}(v_{1}) $ is the degree of vertex $v_{1}$ in $G$.  This paper investigates the extremal values of the Sombor index of chemical trees with exactly one vertex of degree $4$. We characterize the tree attaining the maximum value of the Sombor index and provide the expression for their Sombor indices. Furthermore, we identify the minimum trees and demonstrate that these yield unique tree structures. These results contribute to the structural understanding of degree-based invariants in chemical graph theory. https://comb-opt.azaruniv.ac.ir/article_15165.html This paper introduces and investigates two new sequences, $\{R_{n}^{(k)}\}$ and $\{R_{n}^{(k)}(x)\}$, which provide a distinct generalization of the Mersenne--Lucas numbers and polynomials, respectively, where the index $n$ is expressed in the form $n = sk + r$, with $0 \le r < k$. We derive several identities for these sequences in relation to the classical Mersenne and Mersenne--Lucas numbers and polynomials. Furthermore, we examine their algebraic properties and establish connections with existing sequences and polynomial families. In addition, we obtain closed-form expressions, Cassini-type identities, partial sums, recurrence relations, and various combinatorial identities associated with these sequences. https://comb-opt.azaruniv.ac.ir/article_15166.html An eternal dominating family of a graph $G$ in the eviction game is a collection $\mathcal{D}_{k}=\{D_{1},D_{2},\dots,D_{l}\}$ of dominating sets of $G$ such that (a) $|D_{i}|=|D_{j}|$ for all $i,j\in\{1,2,\dots,l\}$, and (b) for any $i\in \{1,2,\dots,l\}$ and any $v\in D_{i}$, either all neighbours of $v$ belong to $D_{i}$, or there are a neighbour $w$ of $v$ not in $D_{i}$ and an integer $j\in\{1,2,\dots,l\}\setminus\{i\}$ such that $D_{i}\cup\{w\}\setminus \{v\}=D_{j}$. The eviction number of $G$, denoted by $e^{\infty}(G)$, is the smallest cardinality of the sets in such an eternal dominating family.We compare $e^{\infty}$ to the independence number $\alpha$. We show that the ratio $\alpha/e^{\infty}$ is unbounded and construct an infinite class of connected graphs for which $e^{\infty}/\alpha \approx 4/3$. As our main result, we use Ramsey numbers to show that for any integer $k\geq1$, there exists a function $f(k)$ such that any graph with independence number$k$ has eviction number at most $f(k)$. https://comb-opt.azaruniv.ac.ir/article_15167.html In graph theory, the \textit{energy} of a graph \( G \), denoted as \( \mathcal{E}(G) \), is quantified by \(\mathcal{E}(G) = \sum_{i=1}^{n} |\lambda_i|\), where the eigenvalues \( \lambda_1, \lambda_2, \ldots, \lambda_n \) derive from the adjacency matrix of \( G \), and \( n \) represents the vertex total. This research investigates the conditions enabling line graphs of non-regular structures to transform into borderenergetic forms, emphasizing structural traits that drive this transition. The focus includes irregular graphs like the complete bipartite graphs \(K_{a,b}\) across varying \( a \) and \( b \). We also examine corona products, exemplified by \( K_{a,b} \circ K_r \), to identify conditions for the emergence of borderenergetic graphs, thus enhancing comprehension of these graphs through structural and spectral perspectives. https://comb-opt.azaruniv.ac.ir/article_15168.html This paper introduces a novel non-cooperative game played on a capacitated network involving a defender and an attacker. The attacker seeks to maximize the capacity of a path from an origin to a destination, but operates under the constraint of limited network visibility, necessitating a greedy path selection strategy. Conversely, the fully informed defender aims to thwart the attacker's objective by strategically reducing arc capacities within a budgetary constraint. The game unfolds in multiple rounds, with the defender making capacity reduction decisions followed by the attacker's local path extension. This dynamic setting with asymmetric information characterizes the problem as the dynamic maximum capacity path interdiction problem. We present a strongly polynomial algorithm to solve this challenging game. Subsequently, an enhanced algorithm is developed to significantly improve computational efficiency. Extensive computational experiments validate the efficacy of both algorithms, demonstrating that the latter achieves approximately half the runtime of the former. https://comb-opt.azaruniv.ac.ir/article_15169.html In railway stations, there are two track utilization policies: fixed and flexible. Under fixed track utilization, platform tracks are grouped as inbound and outbound parts associated with directions. The trains from the same direction occupy the tracks in each part, and the platforming problem can be considered separately. Under flexible track utilization rule, trains can be assigned to any platform track considering the station layout, which provides more flexibility when platforming. However, it naturally causes a more complex platforming problem. In this paper, we address the train platforming problem in a multi-directional station under flexible track utilization policy. A mixed-integer linear programming formulation is proposed to assign trains to the station’s resources without conflict routes. The objective function is to minimize total weighted delays of trains in which weight refers to the importance level of each train. Previously published studies have only been carried out in small stations with limited trains. However, this study considers busy and complex stations, accommodating more than 1000 trains, and unlike the previous studies, a genetic algorithm is proposed to obtain near-optimal solutions in a short time. Computational analyses are conducted on derived test problems. We construct test problems based on planning periods and traffic volume levels. The performances of the mathematical model and the genetic algorithm are presented in terms of both solution quality and solution time. https://comb-opt.azaruniv.ac.ir/article_15170.html This paper focuses on developing an optimal investment portfolio that combines stocks and related options to minimize unsystematic risk. To achieve this goal, we utilize a deep learning algorithm known as the Deep Galerkin Method (DGM), which accurately prices European call and put options within a long-memory stochastic local volatility framework, thereby enhancing pricing accuracy. Using historical data from 79 stocks in the Russell 2000 Index between 2017 and 2022, we apply various machine learning methods—including Random Forest, K-Nearest Neighbors (KNN), and Support Vector Machines (SVM) to identify optimal portfolios suitable for both risk-seeking and risk-averse investors in 2023. To further refine our investment strategy, we integrate the most effective machine learning methods with a Cardinality-Constrained Mean Variance (CCMV) portfolio optimization model. This integration enables us to construct portfolios tailored to different levels of risk tolerance among investors. https://comb-opt.azaruniv.ac.ir/article_15171.html Cloud computing refers to a paradigm where users request necessary computing resources through the Internet, and now there are numerous cloud service providers offering various resources and services at affordable costs. However, finding a provider that adequately caters to both commercial and operational needs is becoming increasingly challenging. This research proposes a $bi$-objective virtual machine resource allocation problem, which includes payment cost and execution time as the primary objective criteria. The proposed solution approach involves presenting a $bi$-objective mixed integer problem formulation followed by a two-phase method based on combinatorial optimization techniques to discover all Pareto optimal solutions. In phase $1$, the utilized two-phase combinatorial technique locates all supported Pareto optimal solutions, while in phase $2$, it obtains the inner non-supported Pareto optimal solutions. Additionally, suitable weights for payment cost and execution time objectives are determined corresponding to all existing Pareto optimal solutions. https://comb-opt.azaruniv.ac.ir/article_15172.html A graph is called unimodular if the determinant of its adjacency matrix is $\pm1$. Akbari and Kirkland [\textit{Linear Algebra and its Applications}, 421(1):3--15, 2007.] proved that a unicyclic graph is unimodular precisely when it has a unique perfect matching. However, a bicyclic or tricyclic graph with a unique perfect matching need not be unimodular. Recently, Basumatary and Sarma [\textit{Discrete Applied Mathematics}, 346:49-61, 2024.] characterized those bicyclic graphs with a unique perfect matching that are unimodular. In this article, we identify which tricyclic graphs with a unique perfect matching possess the unimodularity property. Our results provide a complete characterization of such graphs. https://comb-opt.azaruniv.ac.ir/article_15173.html Let $G$ be graph with vertex set $V(G)$ and order $n$. A set $S \subseteq V(G)$ is a dominating set of a graph $G$ if every vertex in $V(G) \backslash S$ is adjacent to at least one vertex in $S$. A coalition in a graph $G$ consists of two disjoint sets of vertices $V_1$ and $V_2$,  neither of which is a dominating set but whose union $V_1 \cup V_2$ is a dominating set.  A coalition partition, abbreviated $c$-partition, in a graph $G$ is a vertex partition $\pi=\left\{V_1 , V_2,\dots, V_k\right\}$ such that every set $V_i$ of $\pi$ is either a singleton dominating set, or is not a dominating set but forms a coalition with another set $V_j$ in $\pi$. The sets $V_i$ and $V_j$ are coalition partners in $G$. The coalition number $C(G)$ equals the maximum order $k$ of a $c$-partition of $G$. For any graph $G$ with a $c$-partition $\pi=\left\{V_1,V_2,\dots,V_k\right\}$, the coalition graph $CG(G,\pi)$ of $G$ is a graph with vertex set $V_1,V_2,\dots, V_k$, corresponding one-to-one with the set $\pi$, and two vertices $V_i$ and $V_j$ are adjacent in $CG(G,\pi)$ if and only if the sets $V_i$ and $V_j$ are coalition partners in $\pi$. In [T.W. Haynes, J.T. Hedetniemi, S.T. Hedetniemi, A.A. McRae, and R. Mohan, Coalition graphs, Commun. Comb. Optim. 8 (2023),  423-430], authors proved that for every graph $G$ there exist a graph $H$ and $c$-partition $\pi$ such that $CG(H,\pi)\cong G$, and raised the question: Does there exist a graph $H^*$ of smaller order $n^*$ and size $m^*$ with a $c$-partition $\pi^*$ such that $CG(H^*,\pi^*)\cong G$? In this paper, we constructed a graph $H^*$ of small order and size and a $c$-partition $\pi^*$ such that $CG(H^*,\pi^*)\cong G$.  Recently, Haynes et al. [Introduction to coalitions in graphs, AKCE Int. J. Graphs Comb. 17 (2020), 653-659] defined the coalition count $c(G)$ of a graph $G$ as the maximum number of different coalition in any $c$-partition of $G$.  We characterize all graphs $G$ with $c(G)=1$. Further, imposing some suitable conditions on coalition number, we study the properties of coalition count of graph. https://comb-opt.azaruniv.ac.ir/article_15174.html The superpower graph $\mathcal{S}_\Gamma$ of a finite group $\Gamma$ is an undirected simple graph whose vertices are the elements of the group $\Gamma$, and two distinct vertices $a,b\in \Gamma$ are adjacent if and only if the order of one vertex divides the order of the other vertex, which means that either $o(a)|o(b)$ or $o(b)|o(a)$. In this paper, we investigate the $A_\alpha$-adjacency spectral properties of the superpower graph of groups $D_p\times D_p, D_{p^{k}}, D_{pqr}$, and $D_{p^2q}$, where $p,q,r$ are primes. In particular, we obtain the adjacency, the Laplacian and the signless Laplacian spectra of these graphs, and thereby we prove that the superpower graphs of $D_p\times D_p, $ and $ D_{p^{k}},$ are Laplacain integral. https://comb-opt.azaruniv.ac.ir/article_15176.html A graph $G$ is an efficient closed dominated graph (ECD-graph) if there exists a subset of vertices whose closed neighborhoods partition $V(G)$ and is an efficient open dominated graph (EOD-graph) if there exists a subset of vertices whose open neighborhoods partition $V(G)$. We present a new characterization of ECD- and EOD-graphs that involves independent number and a vertex clique cover of some family of cliques of closed neighborhood graph and open neighborhood graph, respectively, that are intersection graphs of closed and open neighborhoods, respectively. Several consequences are presented as well, one of them with respect to the Vizing's conjecture and the other solves a conjecture on EOD-graphs among toruses $C_t\Box C_r$ posed by Kuziak et al. (Discrete Math. Theoret. Comput. Sci. 16 (2014) 105-120). https://comb-opt.azaruniv.ac.ir/article_15177.html The Maker-Breaker total domination number, $\gamma_{\rm MBT}(G)$, of a graph $G$ is introduced as the minimum number of moves of Dominator to win the Maker-Breaker total domination game, provided that he has a winning strategy and is the first to play. The Staller-start Maker-Breaker total domination number, $\gamma_{\rm MBT}'(G)$, is defined analogously for the game in which Staller starts. Upper and lower bounds on $\gamma_{\rm MBT}(G)$ and on $\gamma_{\rm MBT}'(G)$ are provided and demonstrated to be sharp. It is proved that for any pair of integers $(k,\ell)$ with $2\leq k\leq \ell$, (i) there exists a connected graph $G$ with $\gamma_{\rm MB}(G)=k$ and $\gamma_{\rm MBT}(G)=\ell$, (ii) there exists a connected graph $G'$ with $\gamma_{\rm MB}'(G')=k$ and $\gamma_{\rm MBT}'(G')=\ell$, and (iii) there there exists a connected graph $G''$ with $\gamma_{\rm MBT}(G'')=k$ and $\gamma_{\rm MBT}'(G'')=\ell$. Here, $\gamma_{\rm MB}$ and $\gamma_{\rm MB}'$ are corresponding invariants for the Maker-Breaker domination game. https://comb-opt.azaruniv.ac.ir/article_15178.html Let $G$ be an arbitrary undirected simple connected graph. In this paper, we introduce the modification of the Harary index of $G$ in which the contribution of each edge $uv$ is weighted by $d_u^2 + d_v^2 + d_ud_v$ - a term inspired by the geometry of ellipses - rather than a constant unit weight. Then we compute the values of the Harary-Euler Sombor index of some familiar classes of graphs. Also, we establish mathematical relations between the Harary-Euler Sombor index and other classic indices. Moreover, we state an upper bound for the Harary-Euler Sombor index of bipartite graphs. In addition, we state an upper bound for the Harary-Euler Sombor index of $G$ in terms of the order of $G$ and the largest (smallest) eigenvalue of the Harary-Euler Sombor matrix of $G$ and we introduce a family of graphs for which the given bound is sharp. Finally, we determine the extremum values of the Harary-Euler Sombor index of trees. https://comb-opt.azaruniv.ac.ir/article_15180.html In graph theory, finding the minimal set of vertices with specific covering properties is important. A geodesic transversal of a graph $G$ is a set $S$ of vertices such that every maximal geodesic of $G$ includes at least one vertex from S. The minimum size of a geodesic transversal of $G$ is called geodesic transversal number, denoted as $gt(G)$. It helps in understanding how graphs navigate, how efficiently they communicate, and how to monitor networks. This work finds $gt(G)$ for all graphs with a diameter of 2 and expands the analysis to various classes of graphs with a diameter of 3. https://comb-opt.azaruniv.ac.ir/article_15181.html Given a set $U \subset V$ of vertices in a graph $G = (V, E)$, a {\it private neighbor with respect to the set $U$} is any vertex $w \in V$ having precisely one neighbor, say $v$, in $U$. If $w \in V - U$, then $w$ is called an {\it external private neighbor} of $v$ with respect to $U$. If $w \in U$ then $w$ is called an {\it internal private neighbor} of $v$ with respect to $U$. We also add one special case: if $w \in U$ and $N(w) \cap U = \emptyset$, then we say that $w$ is a {\it self private neighbor} with respect to $U$. By definition, a self private neighbor with respect to $U$ is an isolated vertex in the subgraph of $G$ induced by $U$. In this paper we consider the general problems of trying to find sets of vertices which maximize the number of private neighbors of specific types in a graph. In the process of doing this we define several new maximization parameters of graphs which generalize some known and well-studied parameters of graphs relating to vertex and edge independence, domination and irredundance in graphs. https://comb-opt.azaruniv.ac.ir/article_15183.html A novel vertex-degree-based topological index, namely the Euler Sombor index  was defined as $ EU(G)=\sum\nolimits_{uv\in E(G)}{\sqrt{d_{u}^{2}+d_{v}^{2}+d_{u}d_{v}}}$. Based on this index, here we initiate the distance-based graph index as $\mathcal{E}_{ES}(G)=\sum\nolimits_{uv\in E(G)}{\sqrt{e^2(u)+e^2(v)+e(u)e(v)}}$ and call it the eccentric Euler Sombor index of a (chemical) graph $G = (V(G),E(G))$, where $e(u)$ and $e(v)$ are the eccentricity of $u$ and $v$ in $V(G)$, respectively. We establish basic mathematical properties of this new index. Also, we state some bounds for $\mathcal{E}_{ES}(G)$ in terms of order, size, degrees, radius, and  diameter of $G$. We also determine trees (connected graphs, respectively) of a given order that have the minimum and maximum value of this index. Furthermore, we pose a conjecture about the maximum value of $\mathcal{E}_{ES}(G)$ when $G$ is a connected graph of fixed order. https://comb-opt.azaruniv.ac.ir/article_15184.html Let $G$ be a connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity and remoteness of $G$ are defined as the minimum and maximum, respectively, of the average distances of the vertices of $G$. It was shown by Aouchiche and Hansen [Proximity and remoteness in graphs: bounds and conjectures, Networks 58 no. 2 (2011)] that for a connected graph of order $n$, the difference between remoteness and proximity and the difference between radius and proximity are bounded from above by about $\frac{n}{4}$, and the difference between diameter and proximity is bounded from above by about $\frac{3}{4}n$. In this paper, we show that all three bounds can be improved significantly for simple triangulations (i.e., triangulations), and for graphs of given connectivity. We show that in simple triangulations the above bound on the difference between radius and proximity can be improved to about $\frac{1}{12}n$, and further to about $\frac{1}{16}n$ and $\frac{1}{20}n$ if the graphs is, in addition, $4$-connected or $5$-connected, respectively. Similar improvements are shown for simple quadrangulations (i.e., maximal bipartite planar graphs), and for maximal outerplanar graphs. We further show that the above bound on the difference between remoteness and proximity can be improved to about $\frac{1}{4\kappa}n$ if $G$ is $\kappa$-connected.Finally, we improve the bound on the difference between diameter and proximity to about $\frac{3}{4\kappa}n$ if $G$ is $\kappa$-connected. We present graphs that demonstrate that our bounds are either sharp, or sharp apart from an additive constant, even if restricted to planar graphs.