Communications in Combinatorics and Optimization
https://comb-opt.azaruniv.ac.ir/
Communications in Combinatorics and Optimizationendaily1Sat, 01 Mar 2025 00:00:00 +0330Sat, 01 Mar 2025 00:00:00 +0330On coherent configuration of circular-arc graphs
https://comb-opt.azaruniv.ac.ir/article_14629.html
For any graph, Weisfeiler and &nbsp;Leman assigned the smallest &nbsp;matrix algebra which &nbsp;contains the adjacency matrix of the graph. The coherent configuration underlying this &nbsp;algebra for a graph $\Gamma$ is called the coherent configuration of $\Gamma$, denoted by $\mathcal{X}(\Gamma)$. In this paper, we study the coherent configuration of circular-arc graphs. We give a characterization of the circular-arc graphs $\Gamma$, where $\mathcal{X}(\Gamma)$ &nbsp;is a homogeneous coherent configuration. Moreover, all homogeneous coherent configurations which are obtained in this way are characterized as a subclass of Schurian coherent configurations.On the Sombor Index of Sierpiński and Mycielskian Graphs
https://comb-opt.azaruniv.ac.ir/article_14616.html
In 2020, mathematical chemist, Ivan Gutman, introduced a new vertex-degree-based topological index called the Sombor Index, denoted by $SO(G)$, where $G$ is a simple, connected, finite, graph. This paper aims to present some novel formulas, along with some upper and lower bounds on the Sombor Index of generalized Sierpi\'nski graphs; originally defined by Klav\v{z}ar and Milutinovi\'c by replacing the complete graph appearing in $S(n,k)$ with any graph and exactly replicating the same graph, yielding self-similar graphs of fractal nature; and on the Sombor Index of the $m$-Mycielskian or the generalized Mycielski graph; formed from an interesting construction given by Jan Mycielski (1955); of some simple graphs such as \(K_n\), \(C_n^2\), \(C_n\), and \(P_n\). We also provide Python codes to verify the results for the \(SO\left(S\left(n,K_m\right)\right)\) and \(SO\left(\mu_m\left(K_n\right)\right)\).On the complement of the intersection graph of subgroups of a group
https://comb-opt.azaruniv.ac.ir/article_14612.html
The complement of the intersection graph of subgroups of a group $G$, denoted by $\mathcal{I}^c(G)$, is the graph whose vertex set is the set of all nontrivial proper subgroups of $G$ and its two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K$ is trivial. In this paper, we classify all finite groups whose complement of the intersection graph of subgroups is one of totally disconnected, bipartite, complete bipartite, tree, star graph or $C_3$-free. Also we characterize all the finite groups whose complement of the intersection graph of subgroups is planar.Well ve-covered graphs
https://comb-opt.azaruniv.ac.ir/article_14622.html
A vertex $u$ of a graph $G=(V,E)$ ve-dominates every edge incident to $u$ as well as every edge adjacent to these incident edges. A set $S\subseteq V$ is a vertex-edge dominating set (or a ved-set for short) if every edge of $E$ is ve-dominated by at least one vertex in $S$. A ved-set is independent if its vertices are pairwise non-adjacent. The independent ve-domination number $i_{ve}(G)$ is the minimum cardinality of an independent ved-set and the upper independent ve-domination number $\beta_{ve}(G)$ is the maximum cardinality of a minimal independent ved-set of $G$. In this paper, we are interesting in graphs $G$ such that $i_{ve}(G)=\beta_{ve}(G)$, which we call well ve-covered graphs. We show that recognizing well ve-covered graphs is co-NP-complete, and we present a constructive characterization of well ve-covered trees.Sharp lower bounds on the metric dimension of circulant graphs
https://comb-opt.azaruniv.ac.ir/article_14679.html
For $n \ge 2t+1$ where $t \ge 1$, the circulant graph $C_n (1, 2, \dots , t)$ consists of the vertices $v_0, v_1, v_2, \dots , v_{n-1}$ and the edges $v_i v_{i+1}$, $v_i v_{i+2}, \dots , v_i v_{i + t}$, where $i = 0, 1, 2, \dots , n-1$, and the subscripts are taken modulo $n$. We prove that the metric dimension ${\rm dim} (C_n (1, 2, \dots , t)) \ge \left\lceil \frac{2t}{3} \right\rceil + 1$ for $t \ge 5$, where the equality holds if and only if $t = 5$ and $n = 13$. Thus ${\rm dim} (C_n (1, 2, \dots , t)) \ge \left\lceil \frac{2t}{3} \right\rceil + 2$ for $t \ge 6$. This bound is sharp for every $t \ge 6$.Graphs with unique minimum edge-vertex dominating sets
https://comb-opt.azaruniv.ac.ir/article_14624.html
An edge $e$ of a simple graph $G=(V_{G},E_{G})$ is said to ev-dominate a vertex $v\in V_{G}$ if $e$ is incident with $v$ or $e$ is incident with a vertex adjacent to $v$. A subset $D\subseteq E_{G}$ is an edge-vertex dominating set (or an evd-set for short) of $G$ if every vertex of $G$ is ev-dominated by an edge of $D$. The edge-vertex domination number of $G$ is the minimum cardinality of an evd-set of $G$. In this paper, we initiate the study of the graphs with unique minimum evd-sets that we will call UEVD-graphs. We first present some basic properties of UEVD-graphs, and then we characterize UEVD-trees by equivalent conditions as well as by a constructive method.Lower General Position in Cartesian Products
https://comb-opt.azaruniv.ac.ir/article_14763.html
A subset $S$ of vertices of a graph $G$ is in general position if no shortest path in $G$ contains three vertices of $S$. The general position problem consists of finding the number of vertices in a largest general position set of $G$, whilst the lower general position problem asks for a smallest maximal general position set. In this paper we determine the lower general position numbers of several families of Cartesian products. We also show that the existence of small maximal general position sets in a Cartesian product is connected to a special type of general position set in the factors, which we call a terminal set, for which adding any vertex $u$ from outside the set creates three vertices in a line with $u$ as an endpoint. We give a constructive proof of the existence of terminal sets for graphs with diameter at most three. We also present conjectures on the existence of terminal sets for all graphs and a lower bound on the lower general position number of a Cartesian product in terms of the lower general position numbers of its factors.Commuting graph of an aperiodic Brandt Semigroup
https://comb-opt.azaruniv.ac.ir/article_14627.html
The commuting graph of a finite non-commutative semigroup $S$, denoted by $\Delta(S)$, is the simple graph whose vertices are the non-central elements of $S$ and two distinct vertices $x, y$ are adjacent if $xy = yx$. In this paper, we study the commuting graph of an important class of inverse semigroups viz. Brandt semigroup $B_n$. In this connection, we obtain the automorphism group ${\rm Aut}(\Delta(B_n))$ and the endomorphism monoid End$(\Delta(B_n))$ of $\Delta(B_n)$. We show that ${\rm Aut}(\Delta(B_n)) \cong S_n \times \mathbb{Z}_2$, where $S_n$ is the symmetric group of degree $n$ and $\mathbb{Z}_2$ is the additive group of integers modulo $2$. Further, for $n \geq 4$, we prove that End$(\Delta(B_n)) = $Aut$(\Delta(B_n))$. Moreover, &nbsp;we provide the vertex connectivity and edge connectivity of $\Delta(B_n)$. This paper provides a partial answer to a question posed in \cite{a.Araujo2011} and so we ascertained &nbsp;a class of inverse semigroups whose commuting graph is Hamiltonian.On Zero-Divisor Graph of the ring $\mathbb{F}_p+u\mathbb{F}_p+u^2 \mathbb{F}_p$
https://comb-opt.azaruniv.ac.ir/article_14641.html
In this article, we discussed the zero-divisor graph of a commutative ring with identity $\mathbb{F}_p+u\mathbb{F}_p+u^2 \mathbb{F}_p$ where $u^3=0$ and $p$ is an odd prime. We find the clique number, chromatic number, vertex connectivity, edge connectivity, diameter and girth of a zero-divisor graph associated with the ring. We find some of topological indices and the main parameters of the code derived from the incidence matrix of the zero-divisor graph $\Gamma(R).$ Also, we find the eigenvalues, energy and spectral radius &nbsp;of both adjacency and Laplacian matrices of $\Gamma(R).$Finite Abelian Groups with Isomorphic Inclusion Graphs of Cyclic Subgroups
https://comb-opt.azaruniv.ac.ir/article_14630.html
Let $G$ be a finite group. The directed inclusion graph of cyclic subgroups of $G$, $\overrightarrow{\mathcal{I}_c}(G)$, &nbsp;is the digraph with vertices of all &nbsp;cyclic subgroups of $G$, and for two distinct cyclic subgroups $\langle a \rangle$ and $\langle b \rangle$, there is an arc from $\langle a\rangle $ to $\langle b\rangle $ if and only if $\langle b\rangle \subset \langle a\rangle $. The (undirected ) inclusion graph of cyclic subgroups of $G$, $\mathcal{I}_c(G)$, is the underlying graph of $\overrightarrow{\mathcal{I}_c}(G)$, that is, the vertex set is the set of all cyclic subgroups of $G$ and two distinct cyclic subgroups $\langle a \rangle$ and $\langle b \rangle$ are adjacent if and only if $\langle a\rangle \subset \langle b\rangle$ or $\langle b\rangle \subset \langle a\rangle $. In this paper, we first show that, if $G$ and $H$ are finite groups such that $\mathcal{I}_c(G)\cong \mathcal{I}_c(H)$ and $G$ is cyclic, then $H$ is cyclic. We show that for two cyclic groups $G$ and $H$ of orders $p_1^{\alpha_1} \dots &nbsp;p_t^{\alpha_t}$ and $q_1^{\beta_1} \dots &nbsp;q_s^{\beta_s}$, respectively, $\mathcal{I}_c(G)\cong \mathcal{I}_c(H)$ if and only if $t=s$ and by a suitable $\sigma $, $\alpha_i=\beta_{\sigma (i)}$. Also for any cyclic groups $G,~H$, if $\mathcal{I}_c(G)\cong \mathcal{I}_c(H)$, then $\overrightarrow{\mathcal{I}_c}(G) \cong \overrightarrow{\mathcal{I}_c}(H)$. We also show that for two finite abelian groups $G$ and $H$, $\mathcal{I}_c(G)\cong \mathcal{I}_c(H)$ if and only if $|\pi (G)|=|\pi (H)|$ and by a convenient permutation the graph of their sylow subgroups are isomorphic. In this case, their directed inclusion graphs are isomorphic too.NP-completeness of some generalized hop and step domination parameters in graphs
https://comb-opt.azaruniv.ac.ir/article_14653.html
&lrm;Let $r\geq 2$. A subset $S$ of vertices of a graph $G$ is a $r$-hop independent dominating set if every vertex outside $S$ is at distance $r$ from a vertex of $S$, and for any pair $v, w\in S$, $d(v, w)\neq r$. A $r$-hop Roman dominating function ($r$HRDF) is a function $f$ on $V(G)$ with values $0,1$ and $2$ having the property that for every vertex $v \in V$ with $f(v) = 0$ there is a vertex $u$ with $f(u)=2$ and $d(u,v)=r$. A $r$-step Roman dominating function ($r$SRDF) is a function $f$ on $V(G)$ with values $0,1$ and $2$ having the property that for every vertex $v$ with $f(v)=0$ or $2$, there is a vertex $u$ with $f(u)=2$ and $d(u,v)=r$. A $r$HRDF $f$ is a $r$-hop Roman independent dominating function if for any pair $v, w$ with non-zero labels under $f$, $d(v, w)\neq r$. We show that the decision problem associated with each of $r$-hop independent domination, $r$-hop Roman domination, $r$-hop Roman independent domination and $r$-step Roman domination is NP-complete even when restricted to planar bipartite graphs or planar chordal graphs.Cliques in the extended zero-divisor graph of finite commutative rings
https://comb-opt.azaruniv.ac.ir/article_14651.html
Let $R$ be a finite commutative ring with or without unity and $\Gamma_{e}(R)$ be its extended zero-divisor graph with vertex set $Z^{*}(R)=Z(R)\setminus \lbrace0\rbrace$ and two distinct vertices $x,y$ are adjacent if and only if $x.y=0$ or $x+y\in Z^{*}(R)$. In this paper, we characterize finite commutative rings whose extended zero-divisor graph have clique number $1 ~ \text{or}~ 2$. We completely characterize the rings of the form $R\cong R_1\times R_2 $, where $R_1$ and $R_2$ are local, having clique number $3,~4~\text{or}~5$. Further we determine the rings of the form $R\cong R_1\times R_2 \times R_3$, where $R_1$,$R_2$ and $R_3$ are local rings, to have clique number equal to six.On the distance-transitivity of the folded hypercube
https://comb-opt.azaruniv.ac.ir/article_14656.html
The folded hypercube $FQ_n$ is the Cayley graph Cay$(\mathbb{Z}_2^n,S)$, where $S=\{e_1,e_2,\dots,e_n\}\cup&nbsp;\{u=e_1+e_2+\dots+e_n\}$, and $e_i = (0,\dots, 0, 1, 0,$ $\dots, 0)$, with 1 at the $i$th position, $1\leq i \leq n$. In this paper, we show that the folded hypercube $FQ_n$ is a distance-transitive graph. Then, we study some properties of this graph. In particular, we show that if $n\geq 4$ is an even integer, then the folded hypercube $FQ_n$ is an $automorphic$ graph, that is, $FQ_n$ is a distance-transitive primitive graph which is not a complete or a line graph.Antipodal Number of Cartesian Product of Complete Graphs with Cycles
https://comb-opt.azaruniv.ac.ir/article_14643.html
Let $G$ be a simple connected graph with diameter $d$, and $k\in [1,d]$ be an integer. A radio $k$-coloring of graph $G$ is a mapping $g:V(G)\rightarrow \{0\}\cup \mathbb{N}$ satisfying $\lvert g(u)-g(v)\rvert\geq 1+k-d(u,v)$ for any pair of distinct vertices $u$ and $v$ of the graph $G$, where $d(u,v)$ denotes distance between vertices $u$ and $v$ in $G$. The number ${\text{max}} \{g(u):u\in V(G)\}$ is known as the span of $g$ and is denoted by $rc_k(g)$. The radio $k$-chromatic number of graph $G$, denoted by $rc_k(G)$, is defined as $\text{min} \{rc_k(g) : g \text{ is a radio $k$-coloring of $G$}\}$. For $k=d-1$, the radio $k$-coloring of graph $G$ is called an antipodal coloring. So $rc_{d-1}(G)$ is called the antipodal number of $G$ and is denoted by $ac(G)$. Here, we study antipodal coloring of the Cartesian product of the complete graph $K_r$ and cycle $C_s$, $K_r\square C_s$, for $r\geq 4$ and $s\geq 3$. We determine the antipodal number of $K_r\square C_s$, for even $r\geq 4$ with $s\equiv 1(mod\,4)$; and for any $r\geq 4$ with $s=4t+2$, $t$ odd. Also, for the remaining values of $r$ and $s$, we give lower and upper bounds for $ac(K_r\square C_s)$.The zero-divisor associate graph over a finite commutative ring
https://comb-opt.azaruniv.ac.ir/article_14655.html
In this paper, we introduce the zero-divisor associate graph $\Gamma_D(R)$ over a finite commutative ring $R$. It is a simple undirected graph whose vertex set consists of all non-zero elements of $R$, and two vertices $a, b$ are adjacent if and only if there exist non-zero zero-divisors $z_1, z_2$ in $R$ such that $az_1=bz_2$. We determine the necessary and sufficient conditions for connectedness and completeness of $\Gamma_D(R)$ for a unitary commutative ring $R$. The chromatic number of $\Gamma_D(R)$ is also studied. Next, we characterize the rings $R$ for which $\Gamma_D(R)$ becomes a line graph of some graph. Finally, we give the complete list of graphs with at most 15 vertices which are realizable as $\Gamma_D(R)$, characterizing the associated ring $R$ in each case.Some properties of star-perfect graphs
https://comb-opt.azaruniv.ac.ir/article_14602.html
For a finite simple graph $G=(V, E)$, $\theta_s(G)$ denotes the minimum number of induced stars contained in $G$ such that the union of their vertex sets is $V(G)$, and $ \alpha_s(G)$ denotes the maximum number of vertices in $G$ such that no two are contained in the same induced star of $G$. We call the graph $G$ star-perfect if $\alpha_s(H)=\theta_s(H)$, for every induced subgraph $H$ of $G$. We prove here that no cycle in a star-perfect graph has crossing chords and star-perfect graphs are planar. Also we present a few properties of star perfect graphs.$k$-Secure Sets and $k$-Security Number of a Graph
https://comb-opt.azaruniv.ac.ir/article_14657.html
Let $G=(V, E)$ be a simple connected graph. A nonempty set $S\subseteq V$ is a secure set if every attack on $S$ is defendable. In this paper, $k$-secure sets are introduced as a generalization of secure sets. For any integer $k\geq 0$, a nonempty subset $S$ of $V$ is a $k$-secure set if, for each attack on $S$, there is a defense of $S$ such that for every $v\in S$, the defending set of $v$ contains at least $k$ more elements than that of the attacking set of $v$, whenever the vertex $v$ has neighbors outside $S$. The cardinality of a minimum $k$-secure set in $G$ is the $k$-security number of $G$. Some properties of $k$-secure sets are discussed and a characterization of $k$-secure sets is obtained. Also, 1-security numbers of certain classes of graphs are determined.On spectral properties of neighbourhood second Zagreb matrix of graph
https://comb-opt.azaruniv.ac.ir/article_14659.html
Let $G$ be a simple graph with vertex set $V(G)=\{1,2,\dots,n\}$ and $\delta(i)= \sum\limits_{\{i,j\} \in E(G)}d(j)$, where $d(j)$ is the degree of the vertex $j$ in $G$. Inspired by the second Zagreb matrix and neighborhood first Zagreb matrix of a graph, we introduce the neighborhood second Zagreb matrix of $G$, denoted by $N_F(G)$. It is the $n\times n$ matrix whose $ij$-th entry is equal to $\delta(i)\delta(j)$, if $i$ and $j$ are adjacent in $G$ and $0$, otherwise. The neighborhood second Zagreb spectral radius $\rho_{N_F}(G)$ is the largest eigenvalue of $N_F(G)$. The neighborhood second Zagreb energy $\mathcal{E}(N_F)$ of the graph $G$ is the sum of the absolute values of the eigenvalues of $N_F(G)$. In this paper, we obtain some spectral properties of $N_F(G)$. We provide sharp bounds for $\rho_{N_F}(G)$ and $\mathcal{E}(N_F)$, and obtain the corresponding extremal graphs.Global restrained Roman domination in graphs
https://comb-opt.azaruniv.ac.ir/article_14661.html
A global restrained Roman dominating function on a graph $G=(V,E)$ to be a function $f:V\rightarrow\{0,1,2\}$ such that $f$ is a restrained Roman dominating function of both $G$ and its complement $\overline G$. The weight of a global restrained Roman dominating function is the value $w(f)=\Sigma_{u \in V} f(u)$. The minimum weight of a global restrained Roman dominating function of $G$ is called the global restrained Roman domination number of $G$ and denoted by $\gamma_{grR}(G)$. In this paper we initiate the study of global restrained Roman domination number of graphs. We then prove that the problem of computing $\gamma_{grR}$ is NP-hard even for bipartite and chordal graphs. The global restrained Roman domination of a given graph is studied versus to the other well known domination parameters such as restrained Roman domination number $\gamma_{rR}$ and global domination number $\gamma_g$ by bounding $\gamma_{grR}$ from below and above involving $\gamma_{rR}$ and $\gamma_g$ for general graphs, respectively. We characterize graphs $G$ for which $\gamma_{grR}(G)\in \{1,2,3,4,5\}$. It is shown that: for trees $T$ of order $n$, $\gamma_{grR}(T)=n$ if and only if diameter of $T$ is at most $5$. Finally, the triangle free graphs $G$ for which $\gamma_{grR}(G)=|V|$ are characterized.On the essential dot product graph of a commutative ring
https://comb-opt.azaruniv.ac.ir/article_14662.html
Let $\mathcal{B}$ be a commutative ring with unity $1\neq 0$, $1\leq m &lt;\infty$ be an integer and $\mathcal{R}=\mathcal{B}\times \mathcal{B} \times\cdots\times \mathcal{B}$ ($m$ times). The total essential dot product graph $ETD(\mathcal{R})$ and the essential zero-divisor dot product graph $EZD(\mathcal{R})$ are undirected graphs with the vertex sets $\mathcal{R}^{*} = \mathcal{R}\setminus \{(0,0,...0)\}$ and $Z(\mathcal{R})^*=Z(\mathcal{R})\setminus \{(0,0,...,0)\}$ respectively. Two distinct vertices $w=(w_1,w_2,...,w_m)$ and $z=(z_1,z_2,...,z_m)$ are adjacent if and only if $ann_\mathcal{B}(w\cdot z)$ is an essential ideal of $\mathcal{B}$ (where $w\cdot z=w_1z_1+w_2z_2+\cdots +w_mz_m\in \mathcal{B}$). In this paper, we prove some results on connectedness, diameter and girth of $ETD(\mathcal{R})$ and $EZD(\mathcal{R})$. We classify the ring $\mathcal{R}$ such that $EZD(\mathcal{R})$ and $ETD(\mathcal{R})$ are planar, outerplanar, and of genus one.On Odd-Graceful Coloring of Graphs
https://comb-opt.azaruniv.ac.ir/article_14663.html
For a graph $G(V,E)$ which is undirected, simple, and finite, we denote by $|V|$ and $|E|$ the cardinality of the vertex set $V$ and the edge set $E$ of $G$, respectively. A \textit{graceful labeling} $f$ for the graph $G$ is an injective function ${f}:V\rightarrow \{0,1,2,..., |E|\}$ such that $\{|f(u)-f(v)|:uv\in E\}=\{1,2,...,|E|\}$. A graph that has a graceful-labeling is called \textit{graceful} graph. A vertex (resp. edge) coloring is an assignment of color (positive integer) to every vertex (resp. edge) of $G$ such that any two adjacent vertices (resp. edges) have different colors. A \textit{graceful coloring} of $G$ is a vertex coloring $c: V\rightarrow \{1,2,\ldots, k\},$ for some positive integer $k$, which induces edge coloring $|c(u)-c(v)|$, $uv\in E$. If $c$ also satisfies additional property that every induced edge color is odd, then the coloring $c$ is called an \textit{odd-graceful coloring} of $G$. If an odd-graceful coloring $c$ exists for $G$, then the smallest number $k$ which maintains $c$ as an odd-graceful coloring, is called \textit{odd-graceful chromatic number} for $G$. In the latter case we will denote the odd-graceful chromatic number of $G$ as $\mathcal{X}_{og}(G)=k$. Otherwise, if $G$ does not admit odd-graceful coloring, we will denote its odd-graceful chromatic number as $\mathcal{X}_{og}(G)=\infty$. In this paper, we derived some facts of odd-graceful coloring and determined odd-graceful chromatic numbers of some basic graphs.L(2,1)-labeling of some zero-divisor graphs associated with commutative rings
https://comb-opt.azaruniv.ac.ir/article_14664.html
Let $\mathcal G = (\mathcal V, \mathcal E)$ be a simple graph, an $L(2,1)$-labeling of $\mathcal G$ is an assignment of labels from non-negative integers to vertices of $\mathcal G$ such that adjacent vertices get labels which differ by at least by two, and vertices which are at distance two from each other get different labels. The $\lambda$-number of $\mathcal G$, denoted by $\lambda(\mathcal G)$, is the smallest positive integer $\ell$ such that $\mathcal G$ has an $L(2,1)$-labeling with all labels as &nbsp;members of the set $\{ 0, 1, \dots, \ell \}$. The zero-divisor graph of a finite commutative ring $R$ with unity, denoted by $\Gamma(R)$, is the simple graph whose vertices are all zero divisors of $R$ in which two vertices $u$ and $v$ are adjacent &nbsp;if and only if $uv = 0$ in $R$. In this paper, we investigate $L(2,1)$-labeling of some &nbsp;zero-divisor graphs. We study the \textit{partite truncation}, a graph operation that allows us to obtain a reduced graph of relatively small order from a graph of significantly larger order. We establish the relation between &nbsp;$\lambda$-numbers of the graph &nbsp;and its partite truncated one. We make use of the operation \textit{partite truncation} to contract the zero-divisor graph of a reduced ring to the zero-divisor graph of a Boolean ring.Polycyclic codes over R
https://comb-opt.azaruniv.ac.ir/article_14666.html
In this paper, we &nbsp;discuss the structure of polycyclic codes over the ring $R=\mathbb{F}_q+u\mathbb{F}_q+v\mathbb{F}_q;u^2=\alpha u,v^2=v$ and $uv=vu=0$, where $\alpha$ is an unit element in $R.$ We introduce annihilator self-dual codes, annihilator self-orthogonal codes and annihilator LCD codes over R. Using a Gray map, we define a one to one correspondence between $R$ and $\mathbb{F}_q$ and &nbsp;construct quasi polycyclic &nbsp;codes over the &nbsp;$\mathbb{F}_q$.A Hybrid Conjugate Gradient Method Between MLS and FR in Nonparametric Statistics
https://comb-opt.azaruniv.ac.ir/article_14667.html
This paper proposes a novel hybrid conjugate gradient method for nonparametric statistical inference.The proposed method is a convex combination of the modified linear search (MLS) and Fletcher-Reeves (FR) methods, and it inherits the advantages of both methods. The FR method is known for its fast convergence, while the MLS method is known for itsrobustness to noise. The proposed method combines these advantages to achieve both fast convergence and robustness to noise. Our method is evaluated on a variety of nonparametric statistical problems, including kernel density estimation, regression, and classification. The results show that the new method outperforms the MLS and FR methods in terms of both accuracy and efficiency.A Simple-Intersection Graph of a Ring Approach to Solving Coloring Optimization Problems
https://comb-opt.azaruniv.ac.ir/article_14669.html
In this paper, we introduce a modified version of the simple-intersection graph for semisimple rings, applied to a ring $R$ with unity. The findings from this modified version are subsequently utilized to solve several coloring optimization problems. &nbsp;We demonstrate how the clique number of the simple-intersection graph can be used to determine the maximum number &nbsp;of possibilities that can be selected from a set of $n$ colors without replacement or order, subject to the constraint that &nbsp;any pair shares only one common color. We also show how the domination number can be used to determine the &nbsp;minimum number of possibilities that can be selected, such that any other possibility shares one color with &nbsp;at least one of the selected possibilities, is $n-1$.Optimizing the Gutman Index: A Study of minimum Values Under Transformations of Graphs
https://comb-opt.azaruniv.ac.ir/article_14670.html
The extremal Gutman index is a concept that studies the maximum or minimum value of the Gutman index for a particular class of graphs. This research area is concerned with finding the graphs that have the lowest possible Gutman index within a set of graphs that have been transformed in some way, such as by adding or removing edges or vertices. By understanding the graphs that have the lowest possible Gutman index, researchers can better understand the fundamental principles of graph stability and the role that different graph transformations play in affecting the overall stability of a graph. The research in this area is ongoing and continues to expand as new techniques and algorithms are developed. The findings from this research have the potential to have a significant impact on a wide range of fields and can lead to new and more effective ways of analyzing and understanding complex systems and relationships in a variety of applications. This paper focuses on the study of specific types of trees that are defined by fixed parameters and characterized based on their Gutman index. Specifically, we explore the structural properties of graphs that have the lowest Gutman index within these classes of trees. To achieve this, we utilize various graph transformations that either decrease or increase the Gutman index. By applying these transformations, we construct trees that satisfy the desired criteria.Injective coloring of generalized Mycielskian of graphs
https://comb-opt.azaruniv.ac.ir/article_14671.html
The injective chromatic number $\chi_i(G)$ of a graph $G$ is the smallest number of colors required to color the vertices of $G$ such that any two vertices with a common neighbor are assigned distinct colors. The Mycielskian or Mycielski graph $\mu(G)$ of a graph $G$, introduced by Jan Mycielski in 1955 has the property that, these graphs have large chromatic number with small clique number. The generalized Mycielskian $\mu_m(G),m&gt;0$ (also known as cones over graphs) are the natural generalizations of the Mycielski graphs. In this paper, sharp bounds are obtained for the injective chromatic number of generalized Mycielskian of any graph $G$. Further, the injective chromatic number of generalized Mycielskian of some special classes of graphs such as paths, cycles, complete graphs, and complete bipartite graphs are obtained.Degree distance index of class of graphs
https://comb-opt.azaruniv.ac.ir/article_14673.html
The topological indices are the numerical parameters of a graph that characterize the topology of a graph and are usually graph invariant. The topological indices are classified based on the properties of graphs. The degree distance index is the topological index which is calculated by counting the degrees and distance between the vertices. In this paper, the degree distance index of the connected thorn graph, the graph obtained by joining an edge between two connected graphs, and one vertex union of two connected graphs are calculated.Vertex-degree function index on tournaments
https://comb-opt.azaruniv.ac.ir/article_14674.html
Let $G$ be a simple graph with vertex set $V=V(G)$ and edge set $E=E(G)$. For a real function $f$ defined on nonnegative real numbers, the vertex-degree function index $H_{f}(G)$ is defined as$$H_{f}(G)=\sum_{u\in V(G)}f(d_{u}).$$In this paper we introduce the vertex-degree function index $H_{f}(D)$ of a digraph $D$. After giving some examples and basic properties of $H_{f}(D)$, we find the extremal values of $H_{f}$ among all tournaments with a fixed number of vertices, when $f$ is a continuous and convex (or concave) real function on $\left[ 0,+\infty \right)$.The First Leap Zagreb Coindex of Some Graph Operations
https://comb-opt.azaruniv.ac.ir/article_14675.html
In the last years, Naji et al. have introduced leap Zagreb indices conceived depending on the second degrees of vertices, where the second degree of a vertex $v$ in a graph $G$ is equal to the number of its second neighbors and denoted by $d_2(v/G)$. &nbsp;Analogously, the leap Zagreb coindices were introduced by Ferdose and Shivashankara. The first leap Zagreb coindex of a graph is defined as &nbsp;$\overline{L_1}(G)=\sum_{uv\not\in E_2(G)}(d_2(u)+d_2(v))$, where $E_2(G)$ is the 2-distance (second) edge set of $G$, In this paper, we present explicit exact expressions for the first leap Zagreb coindex $\overline{L_1}(G)$ of some graph operations.Intuitionistic fuzzy Sombor indices: A novel approach for improving the performance of vaccination centers
https://comb-opt.azaruniv.ac.ir/article_14676.html
Intuitionistic fuzzy graphs are generalizations of fuzzy graphs, in which each vertex is assigned an ordered pair whose first coordinate gives the membership value and the second coordinate gives the non-membership value. There are many theoretical parameters to study different types of graphs and fuzzy graphs, topological indices are one of them. Sombor indices are important in explaining the topology of a graph, and were found to possess useful applicative properties. The two versions of the Sombor indices ($SO_3$ and $SO_4$)are converted into an intuitionistic fuzzy framework, and then formulas for different kinds of graphs are calculated. Our study also involves setting up a network of vaccination centers during a pandemic and applying intuitionistic fuzzy-based topological indices to assess their performance. With the help of this application, we highlight the practical implication and benefits of employing intuitionistic fuzzy-based techniques in vaccination centers. Through a comparative analysis, we evaluate which index is more efficient.On the Zero Forcing Number of Complementary Prism Graphs
https://comb-opt.azaruniv.ac.ir/article_14677.html
The zero forcing number of a graph is the minimum cardinality among all the zero forcing sets of a graph $G$. &nbsp;The aim of this article is to compute the zero forcing number of complementary prism graphs. &nbsp;Some bounds on the zero forcing number of complementary prism graphs are presented. The remainder of this article discusses the following result. &nbsp;Let $G$ and $\overline{G }$ be connected graphs. Then $Z(G\overline{G})\leq n-1$ if and only if &nbsp;there exists two vertices $v_i,v_j \in V(G)$ and $i\neq j$ such that, either $N(v_i) \subseteq N(v_j)$ or $N[v_i] \subseteq N[v_j]$ in $G$.A characterization of locating Roman domination edge critical graphs
https://comb-opt.azaruniv.ac.ir/article_14678.html
A Roman dominating function (or just \textit{RDF}) on a graph $G =(V, E)$ is a function $f: V \longrightarrow \{0, 1, 2\}$ 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 weight of an \textit{RDF} $f$ is the value $f(V)=\sum_{u \in {V}}f(u)$. An \textit{RDF} $f$ can be represented as $f=(V_0,V_1,V_2)$, where $V_i=\{v\in V:f(v)=i\}$ for $i=0,1,2$. An \textit{RDF} $f=(V_0,V_1,V_2)$ is called a locating Roman dominating function (or just \textit{L\textit{RDF}}) if $N(u)\cap V_2\neq N(v)\cap V_2$ for any pair $u,v$ of distinct vertices of $V_0$. The locating-Roman domination number $\gamma_R^L(G)$ is the minimum weight of an \textit{L\textit{RDF}} of $G$. A graph $G$ is said to be a locating Roman domination edge &nbsp;critical graph, or just $\gamma_R^L$-edge critical graph, if $\gamma_R^L(G-e)&gt;\gamma_R^L(G)$ for all $e\in E$. The purpose of this paper is to characterize the class of $\gamma_R^L$-edge critical graphs.On the ordering of the Randić index of unicyclic and bicyclic graphs
https://comb-opt.azaruniv.ac.ir/article_14680.html
Let $d_x$ be the degree of the vertex $x$ in a graph $G$. The Randić index of $G$ is defined by $R(G) = \sum_{xy \in E(G)} (d_x d_y)^ {-\frac{1}{2}}$. Recently, Hasni et al. [Unicyclic graphs with Maximum Randi\'{c} indices, Communication in Combinatorics and Optimization, 1 (2023), 161--172] obtained the ninth to thirteenth maximum Randić indices among the unicyclic graphs with $n$ vertices. In this paper, we correct the ordering of Randić index of unicyclic graphs. In addition, we present the ordering of maximum Randi\'c index among bicyclic graphs of order $n$.A note on the re-defined third Zagreb index of trees
https://comb-opt.azaruniv.ac.ir/article_14681.html
For a graph $\Gamma$&lrm;, &lrm;the re-defined third Zagreb index is defined as $$ReZG_3(\Gamma)=\sum_{ab\in E(\Gamma)}\deg_\Gamma(a) &lrm;\deg_\Gamma(b)\Big(&lrm;\deg_\Gamma(a)+&lrm;\deg_\Gamma(b)\Big)&lrm;&lrm;,$$&lrm;&lrm;where $\deg_\Gamma(a)$ is the degree of&lrm; &lrm;vertex $a$&lrm;. &lrm;We prove for any tree $T$ with $n$ vertices and maximum degree $\Delta$&lrm;, &lrm;&lrm;$ReZG_3(T)\geq&lrm;16n+\Delta^3+2\Delta^2-13\Delta-26$ &lrm;when &lrm;&lrm;$&lrm;\Delta&lt; n-1&lrm;$ &lrm;and&lrm;&nbsp;$ReZG_3(T)=&lrm;n\Delta^2+n\Delta-\Delta^2-\Delta$ &lrm;when &lrm;&lrm;$&lrm;\Delta=n-1&lrm;$.&nbsp;&lrm;Also we determine the corresponding extremal trees&lrm;. &lrm;&lrm;A new construction of regular and quasi-regular self-complementary graphs
https://comb-opt.azaruniv.ac.ir/article_14684.html
A graph $G$ with a vertex set $V$ and an edge set $E$ is called regular if the degree of every vertex is the same. A quasi-regular graph is a graph whose vertices have one of two degrees $r$ and $r-1$, for some positive integer $r$. A graph $G$ is said to be self-complementary if $G$ is isomorphic to it's complement $\overline{G}$. In this paper we give a new method for construction of regular and quasi-regular self-complementary graph.A Short Note on Double Roman Domination in Graphs
https://comb-opt.azaruniv.ac.ir/article_14685.html
In this short note, we report an erroneous result of Mojdeh, Parsian and Masoumi relating the double Roman domination number to the enclaveless number and the differential of a graph. &lrm; &lrm; &lrm; &lrm; &lrm; &lrm; &lrm; &lrm; &lrm; &lrm; &lrm; &lrm; &lrm; &lrm; &lrm; &lrm; &lrm; &lrm; &lrm; &lrm;Skew cyclic codes over $\mathbb{Z}_4+v\mathbb{Z}_4$ with derivation: structural properties and computational results
https://comb-opt.azaruniv.ac.ir/article_14686.html
In this work, we study a class of skew cyclic codes over the ring $R:=\mathbb{Z}_4+v\mathbb{Z}_4,$ where $v^2=v,$ with an automorphism $\theta$ and a derivation $\Delta_\theta,$ namely codes as modules over a skew polynomial ring $R[x;\theta,\Delta_{\theta}],$ whose multiplication is defined using an automorphism $\theta$ and a derivation $\Delta_{\theta}.$ We investigate the structures of a skew polynomial ring $R[x;\theta,\Delta_{\theta}].$ We define $\Delta_{\theta}$-cyclic codes as a generalization of the notion of cyclic codes. The properties of $\Delta_{\theta}$-cyclic codes as well as dual $\Delta_{\theta}$-cyclic codes are derived. As an application, some new linear codes over $\mathbb{Z}_4$ with good parameters are obtained by Plotkin sum construction, also via a Gray map as well as residue and torsion codes of these codes.On distance Laplacian spectral invariants of brooms and their complements
https://comb-opt.azaruniv.ac.ir/article_14687.html
For a connected graph $G$ of order $n$, the distance Laplacian matrix $D^L(G)$ is defined as $D^L(G)=Tr(G)-D(G)$, where $Tr(G)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. The largest eigenvalue of $D^L(G)$ is the distance Laplacian spectral radius of $G$ and the quantity $DLE(G)=\sum\limits_{i=1}^{n}|\rho^L_i(G)-\frac{2W(G)}{n}|$, where $W(G)$ is the Wiener index of $G$, is the distance Laplacian energy of $G$. Brooms of diameter $4$ are the trees obtained from the path $P_{5}$ by appending pendent vertices at some vertex of $ P_{5}$. One of the interesting and important problems in spectral graph theory is to find extremal graphs for a spectral graph invariant and ordering them according to this graph invariant. This problem has been considered for many families of graphs with respect to different graph matrices. In the present article, we consider this problem for brooms of diameter $4$ and their complements with respect to their distance Laplacian matrix. Formally, we discuss the distance Laplacian spectrum and the distance Laplacian energy of brooms of diameter $4$. We will prove that these families of trees can be ordered in terms of their distance Laplacian energy and the distance Laplacian spectral radius. Further, we obtain the distance Laplacian spectrum and the distance Laplacian energy of complement of the family of double brooms and order them in terms of the smallest non-zero distance Laplacian eigenvalue and the distance Laplacian energy.Restrained double Roman domatic number
https://comb-opt.azaruniv.ac.ir/article_14688.html
Let $G$ be a graph with vertex set $V(G)$. A double Roman dominating function (DRDF) on a graph $G$ is a function $f:V(G)\longrightarrow\{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least two neighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ mus have at least one neighbor $u$ with $f(u)\ge 2$. If $f$ is a DRDF on $G$, then let $V_0=\{v\in V(G): f(v)=0\}$. A restrained double Roman dominating function is a DRDF $f$ having the property that the subgraph induced by $V_0$ does not have an isolated vertex. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct restrained double Roman dominating functions on $G$ with the property that $\sum_{i=1}^df_i(v)\le 3$ for each $v\in V(G)$ is called a restrained double Roman dominating family (of functions) on $G$. The maximum number of functions in a restrained double Roman dominating family on $G$ is the restrained double Roman domatic number of $G$, denoted by $d_{rdR}(G)$. We initiate the study of the restrained double Roman domatic number, and we present different sharp bounds on $d_{rdR}(G)$. In addition, we determine this parameter for some classes of graphs.Some results on the complete sigraphs with exactly three non-negative eigenvalues
https://comb-opt.azaruniv.ac.ir/article_14689.html
Let $(K_{n},H^-)$ be a complete sigraph of order $n$ whose negative edges induce a subgraph $H$. In this paper, we characterize $(K_n,H^-)$ with exactly 3 non-negative eigenvalues, where $H$ is a non-spanning two-cyclic subgraph of $K_n$.A modified public key cryptography based on generalized Lucas matrices
https://comb-opt.azaruniv.ac.ir/article_14690.html
In this paper, we propose a generalized Lucas matrix (a recursive matrix of higher order) obtained from the generalized Fibonacci sequences. We obtain their algebraic properties such as direct inverse calculation, recursive nature, etc. Then, we propose a modified public key cryptography using the generalized Lucas matrices as a key element that optimizes the keyspace construction complexity. Furthermore, we establish a key agreement for encryption-decryption with a combination of the terms of generalized Lucas sequences under the residue operation.Total coalitions of cubic graphs of order at most 10
https://comb-opt.azaruniv.ac.ir/article_14694.html
A total coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a total dominating set but whose union $V_{1}\cup V_{2}$, is a total dominating set. A total coalition partition in a&nbsp;graph $G$ of order $n=|V|$ is a vertex partition $\tau = \{V_1, V_2, \dots , V_k \}$ such that every set $V_i \in \tau$ is not a total dominating set but forms a total coalition with another set $V_j\in \tau$ which is not a total dominating set.&nbsp;The total coalition number $TC(G)$ equals the maximum $k$ of a total coalition partition of $G$. In this paper, we determine the total coalition number of all cubic graphs of order $n \le 10$.Erratum to the paper ``A study on graph topology'' (Published in Commun. Comb. Optim. 8 (2023), no. 2, 397-409.)
https://comb-opt.azaruniv.ac.ir/article_14695.html
In this paper, we will point out errors in Theorem 2, Theorem 4, Theorem 5, Proposition 2, Proposition 3, Theorem 8, and Theorem 9 &nbsp;by giving suitable counterexamples. The statements of Theorem 2, Theorem 5, Proposition 2 and Proposition 3 of this paper have been reformulated and proofs are given.On Co-Maximal Subgroup Graph of $D_n$
https://comb-opt.azaruniv.ac.ir/article_14696.html
Let $G$ be a group and $S$ be the collection of all non-trivial proper subgroups of $G$. The co-maximal subgroup graph $\Gamma(G)$ of a group $G$ is defined to be a graph with $S$ as the set of vertices and two distinct vertices $H$ and $K$ are adjacent if and only if $HK=G$. In this paper, we study the comaximal subgroup graph on finite dihedral groups. In particular, we study order, maximum and minimum degree, diameter, girth, domination number, chromatic number and perfectness of comaximal subgroup graph of dihedral groups. Moreover, we prove some isomorphism results on comaximal subgroup graph of dihedral groups.Leavitt path algebras for order prime Cayley graphs of finite groups
https://comb-opt.azaruniv.ac.ir/article_14697.html
In this paper, we generalize the concept of Cayley graphs associated to finite groups. The aim of this paper is the characterization of graph theoretic properties of new type of directed graph $\Gamma_P(G;S)$ and algebraic properties of Leavitt path algebra of order prime Cayley graph $O\Gamma(G;S)$, where $G$ is a finite group with a generating set $S$. We show that the Leavitt path algebra of order prime Cayley graph $L_K(O\Gamma(G;S))$ of a non trivial finite group $G$ with any generating set $S$ over a field $K$ is a purely infinite simple ring. Finally, we prove that the Grothendieck group of the Leavitt path algebra $L_K(\Gamma_P(D_n;S))$ is isomorphic to $\mathbb{Z}_{2n-1}$, where $D_n$ is the dihedral group of degree $n$ and $S=\left\{a, b\right\}$ is the generating set of $D_n$.On Connected Bipartite $Q$-Integral Graphs
https://comb-opt.azaruniv.ac.ir/article_14698.html
A graph $G$ is said to be $H$-free if $G$ does not contain $H$ as an induced subgraph. Let $\mathcal{S}_{n}^2(m)$ be a \textit{variation of double star $\mathcal{S}_{n}^2$} obtained by adding m (&lt;=n) disjoint edges between the pendant vertices which are at distance 3 in $\mathcal{S}_{n}^2$. A graph having integer eigenvalues for its signless Laplacian matrix is known as a Q-integral graph. The Q-spectral radius of a graph is the largest eigenvalue of its signless Laplacian. Any connected Q-integral graph G with Q-spectral radius 7 and maximum edge-degree 8 is either $K_{1,4}\square K_2$ or contains $\mathcal{S}_{4}^2(0)$ as an induced subgraph or is a bipartite graph having at least one of the induced subgraphs $\mathcal{S}_{4}^2(m)$, (m=1, 2, 3). In this article, we improve this result by showing that every connected Q-integral graph G having Q-spectral radius 7, maximum edge-degree 8 is always bipartite and $\mathcal{S}_{4}^2(3)$-free.Total Roman Domination and Total Domination in Unit Disk Graphs
https://comb-opt.azaruniv.ac.ir/article_14699.html
Let $G=(V,E)$ be a simple, undirected and connected graph. A Roman dominating function (RDF) on the graph $G$ is a function $f:V\rightarrow\{0,1,2\}$ such that each vertex $v\in V$ with $f(v)=0$ is adjacent to at least one vertex $u\in V$ with $f(u)=2$. A total Roman dominating function (TRDF) of $G$ is a function $f:V\rightarrow\{0,1,2\}$ such that $(i)$ it is a Roman dominating function, and &nbsp;$(ii)$ the vertices with non-zero weights induce a subgraph with no isolated vertex. The total Roman dominating set (TRDS) problem is to minimize the associated weight, $f(V)=\sum_{u\in V} f(u)$, called the total Roman domination number ($\gamma_{tR}(G)$). Similarly, a subset $S\subseteq V$ is said to be a total dominating set (TDS) on the graph $G$ if $(i)$ $S$ is a dominating set of $G$, and $(ii)$ &nbsp;the induced subgraph $G[S]$ does not have any isolated vertex. The objective of the TDS problem is to minimize the cardinality of the TDS of a given graph. The TDS problem is NP-complete for general graphs. &nbsp;In this paper, we propose a simple $10.5\operatorname{-}$factor approximation algorithm for TRDS problem in UDGs. The running time of the proposed algorithm is $O(|V|\log k)$, where $k$ is the number of vertices with weights $2$. It is an improvement over the best-known $12$-factor approximation algorithm with running time $O(|V|\log k)$ available in the literature. Next, we propose another algorithm for the TDS problem in UDGs, which improves the previously best-known approximation factor from $8$ to $7.79$. The running time of the proposed algorithm is $O(|V|+|E|)$.On e-Super (a, d)-Edge Antimagic Total Labeling of Total Graphs of Paths and Cycles
https://comb-opt.azaruniv.ac.ir/article_14701.html
A $(p, q)$-graph $G$ is &nbsp;{\it $(a, d)$-edge antimagic total} if there exists a bijection $f$ from $V(G) \cup E(G)$ to $\{1, 2, \dots, p+q\}$ such that for each edge $uv \in E(G)$, the edge weight $\Lambda(uv) = f(u) + f(uv) + f(v)$ forms an arithmetic progression with first term $a &gt; 0$ and common difference $d \geq 0$. An $(a, d)$-edge antimagic total labeling in which the vertex labels are $1, 2, \dots, p$ and edge labels are $p+1, p+2, \dots, p+q$ is called a {\it super} $(a, d)$-{\it edge antimagic total labeling}. Another variant of $(a, d)$-edge antimagic total labeling called as e-super $(a, d)$-edge antimagic total labeling in which the edge labels are $1, 2, \dots, q$ and vertex labels are $q+1, q+2, \dots, q+p$. In this paper, we investigate the &nbsp;existence of e-super $(a, d)$-edge antimagic total labeling for total graphs of paths, copies of cycles and disjoint union of cycles.Strong $k$-transitive oriented graphs with large minimum degree
https://comb-opt.azaruniv.ac.ir/article_14706.html
A digraph $D=(V,E)$ is $k$-transitive if for any directed $uv$-path of length $k$, we have $(u,v) \in E$. In this paper, we study the structure of strong $k$-transitive oriented graphs having large minimum in- or out-degree. We show that such oriented graphs are \emph{extended cycles}. As a consequence, we prove that Seymour's Second Neighborhood Conjecture (SSNC) holds for $k$-transitive oriented graphs for $k \leq 11$. Also we confirm Bermond--Thomassen Conjecture for $k$-transitive oriented graphs for $k \leq 11$. A characterization of $k$-transitive oriented graphs having a hamiltonian cycle for $k \leq 6$ is obtained immediately.2-semi equivelar maps on the torus and the Klein bottle with few vertices
https://comb-opt.azaruniv.ac.ir/article_14707.html
The $k$-semi equivelar maps, for $k \geq 2$, are generalizations of maps on the surfaces of Johnson solids to closed surfaces other than the 2-sphere. In the present study, we determine 2-semi equivelar maps of curvature 0 exhaustively on the torus and the Klein bottle. Furthermore, we classify (up to isomorphism) all these 2-semi equivelar maps on the surfaces with up to 12 vertices.Exploring the Precise Edge Irregularity Strength of Generalized Arithmetic and Geometric Staircase Graphs
https://comb-opt.azaruniv.ac.ir/article_14708.html
In the context of a finite undirected graph $\zeta$, an edge irregular labelling is defined as a labelling of its vertices with some labels in such a way that each edge has a unique weight, which is determined by the sum of the labels of its endpoints. The main objective of this study is to determine the smallest positive integer $n$ for which it is possible to assign a total edge irregular labelling to $\zeta$ with $n$ as the biggest label. This investigation focuses particularly on cases where $\zeta$ represents the generalized arithmetic and generalized geometric staircase graphs. Within this paper, the definition of generalized geometric staircase graph is proposed. Moreover, we not only establish the edge irregularity strength of these two kind of graphs but also present a method for creating the corresponding edge irregular labelling.A hybrid branch-and-bound and interior-point algorithm for stochastic mixed-integer nonlinear second-order cone programming
https://comb-opt.azaruniv.ac.ir/article_14714.html
One of the chief attractions of stochastic mixed-integer second-order cone programming is its diverse applications, especially in engineering (Alzalg and Alioui, {\em IEEE Access}, 10:3522-3547, 2022). The linear and nonlinear versions of this class of optimization problems are still unsolved yet. In this paper, we develop a hybrid optimization algorithm coupling branch-and-bound and primal-dual interior-point methods for solving two-stage stochastic mixed-integer nonlinear second-order cone programming. The adopted approach uses a branch-and-bound technique to handle the integer variables and an infeasible interior-point method to solve continuous relaxations of the resulting subproblems. The proposed hybrid algorithm is also implemented to data to show its efficiency.Global Malmquist productivity index for evaluation of multistage series systems with undesirable and non-discretionary data
https://comb-opt.azaruniv.ac.ir/article_14718.html
Data Envelopment Analysis measures relative efficiency, in which the performances of the DMUs in a group are compared. In this approach, an efficient unit in one group may be considered inefficient compared to the units of other groups and vice versa. To solve this weakness, two known productivity indexes, the Malmquist and Luenberger, have been introduced to evaluate units (or systems) from one period to another. The existence of special types of data such as undesirable and non-discretionary in some multi-stage series systems is unavoidable. The evaluation of such systems inthe simultaneous presence of the mentioned data and different periods has not been done so far. Therefore, in this study, we have presented a model with a new approach to evaluate them. At the end of the study, we checked the proposed model&rsquo;s ability by providing comparative and structural examples. We have shown that without undesirable and non-discretionary data, the proposed is better than other models. Also, this model has been used for the first time and obtained acceptable results in the presence of these data.Nonlinear inclusion for thermo-electro-elastic: existence, dependence and optimal control
https://comb-opt.azaruniv.ac.ir/article_14719.html
The objective of this paper is to examine a model of a thermo-electro-elastic body situated on a semi-insulator foundation. Friction is characterized by Tresca's friction law, and the contact is bilateral. The primary contribution is to derive the weak variational formulation of the model, constituting a system that couples three inclusions where the unknowns are the strain field, the electric field, and the temperature field. Subsequently, we demonstrate the unique solvability of the system, along with the continuous dependence of its solution under consideration. The secondary contribution involves the investigation of an associated optimal control problem, for which we establish the existence and convergence results. The proofs rely on arguments related to monotonicity, compactness, convex analysis, and lower semicontinuity.On the strength and independence number of powers of paths and cycles
https://comb-opt.azaruniv.ac.ir/article_14720.html
A numbering $f$ of a graph $G$ of order $n$ is a labeling that assigns distinct elements of the set $\left\{1,2, \ldots, n \right\}$ to the vertices of $G$. The strength $\mathrm{str}\left(G\right) $ of $G$ is defined by $\mathrm{str}\left( G\right) =\min \left\{ \mathrm{str}_{f}\left( G\right)\left\vert f\text{ is a numbering of }G\right. \right\}$, where $\mathrm{str}_{f}\left( G\right) =\max \left\{ f\left( u\right)+f\left( v\right) \left\vert uv\in E\left( G\right) \right. \right\} $.Using the concept of independence number of a graph, we determine formulas for the strength of powers of paths and cycles. To achieve the latter result, we establish a sharp upper bound for the strength of a graph in terms of its order and independence number and a formula for the independence number of powers of cycles.Complete solutions on local antimagic chromatic number of three families of disconnected graphs
https://comb-opt.azaruniv.ac.ir/article_14722.html
An edge labeling of a graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we study local antimagic labeling of disjoint unions of stars, paths and cycles whose components need not be identical. Consequently, we completely determined the local antimagic chromatic numbers of disjoint union of 2 stars, paths, and 2-regular graphs with at most one odd order component respectively.Triangular type-2 fuzzy goal programming approach for bimatrix games
https://comb-opt.azaruniv.ac.ir/article_14726.html
This paper addresses a bimatrix game in which the satisfactory degrees of the players are vague. Type-2 fuzzy goal programming technique is used to describe the game. Then, the notion of equilibrium points is introduced and an optimization problem is given to calculate them. Moreover, the special case that the type-2 fuzzy goals are triangular is investigated. Finally, an applicable example is presented to illustrate the results.The crossing numbers of join products of $K_4\cup K_1$ with cycles
https://comb-opt.azaruniv.ac.ir/article_14732.html
The crossing number $\mathrm{cr}(G)$ of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. In the paper, we extend known results concerning crossing numbers of join products of two small graphs with cycles. The crossing number of the join product $G^\ast + C_n$ for the disconnected graph $G^\ast$ consisting of the complete graph $K_{4}$ and one isolated vertex is given, where $C_n$ is the cycle on $n$ vertices. The proof of the main result is done with the help of lemma whose proof is based on a special redrawing technique. Up to now, the crossing numbers of $G + C_n$ were done only for a few disconnected graphs $G$. Finally, by adding new edge to the graph $G^\ast$, we are able to obtain the crossing number of $G_1+C_n$ for one other graph $G_1$ of order five.A New Measure for Transmission Irregularity Extent of Graphs
https://comb-opt.azaruniv.ac.ir/article_14733.html
The transmission of a vertex ${\varsigma}$ in a connected graph $\mathcal{J}$ is the sum of distances between ${\varsigma}$ and all other vertices of $\mathcal{J}$. A graph $\mathcal{J}$ is called transmission regular if all vertices have the same transmission. In this paper, we propose a new graph invariant for measuring the transmission irregularity extent of transmission irregular graphs. This invariant which we call the total transmission irregularity number (TTI number for short) is defined as the sum of the absolute values of the difference of the vertex transmissions over all unordered vertex pairs of a graph. We investigate some lower and upper bounds on the TTI number which reveal its connection to a number of already established indices. In addition, we compute the TTI number for various families of composite graphs and for some chemical graphs and nanostructures derived from them.Maximal outerplanar graphs with semipaired domination number double the domination number
https://comb-opt.azaruniv.ac.ir/article_14746.html
A subset $S$ of vertices in a graph $G$ is a dominating set if every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. If the graph $G$ has no isolated vertex, then a pair dominating set $S$ of $G$ is a dominating set of $G$ such that $G[S]$ has a perfect matching. Further, a semipaired dominating set of $G$ is a dominating set of $G$ with the additional property that the set $S$ can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The domination number $\gamma(G)$ is the minimum cardinality of a dominating set of $G$. Similarly, the paired (semipaired) domination number $\gamma_{pr}(G)$ $(\gamma_{pr2}(G))$ is the minimum cardinality of a paired (semipaired) dominating set of $G$. It is known that for a graph $G$, $\gamma(G) \le \gamma_{pr2}(G) \le \gamma_{pr}(G) \le 2\gamma(G)$. In this paper, we characterize maximal outerplanar graphs $G$ satisfying $\gamma_{pr2}(G) = 2\gamma(G)$. Hence, our result yields the characterization of maximal outerplanar graphs $G$ satisfying $\gamma_{pr}(G) = 2\gamma(G)$.Set colorings of the Cartesian product of some graph families
https://comb-opt.azaruniv.ac.ir/article_14748.html
Neighbor-distinguishing colorings, which are colorings that induce a proper vertex coloring of a graph, have been the focus of different studies in graph theory. One such coloring is the set coloring. For a nontrivial graph $G$, let $c:V(G)\to \mathbb{N}$ and define the neighborhood color set $NC(v)$ of each vertex $v$ as the set containing the colors of all neighbors of $v$. The coloring $c$ is called a set coloring if $NC(u)\neq NC(v)$ for every pair of adjacent vertices $u$ and $v$ of $G$. The minimum number of colors required in a set coloring is called the set chromatic number of $G$ and is denoted by $\chi_s (G)$. In recent years, set colorings have been studied with respect to different graph operations such as join, comb product, middle graph, and total graph. Continuing the theme of these previous works, we aim to investigate set colorings of the Cartesian product of graphs. In this work, we investigate the gap given by $\max\{ \chi_s(G), \chi_s(H) \} - \chi_s(G\ \square\ H)$ for graphs $G$ and $H$. In relation to this objective, we determine the set chromatic numbers of the Cartesian product of some graph families.Elliptic Sombor index of chemical graphs
https://comb-opt.azaruniv.ac.ir/article_14751.html
Let $G$ be a simple graph. The elliptic Sombor index of $G$ is defined as$$&nbsp; &nbsp; ESO(G) = \sum_{uv} \left(d_{u}+ d_{v} \right)\sqrt{d^{2}_{u}+d^{2}_{v}},$$&nbsp;where $d_{u}$ denotes the degree of the vertex $u$, and the sum runs over the set of edges of $G$. In this paper we solve the extremal value problem of $ESO$ over the set of (connected) chemical graphs and over the set of chemical trees, with equal number of vertices.γ-Total Dominating Graphs of Lollipop, Umbrella, and Coconut Graphs
https://comb-opt.azaruniv.ac.ir/article_14753.html
A total dominating set of a graph $G$ is a set $D\subseteq V(G)$ such that every vertex of $G$ is adjacent to some vertex in $D$. The total domination number $\gamma_{t}(G)$ of $G$ is the minimum cardinality of a total dominating set. The $\gamma$-total dominating graph $TD_{\gamma}(G)$ of $G$ is the graph whose vertices are minimum total dominating sets, and two minimum total dominating sets of $TD_{\gamma}(G)$ are adjacent if they differ by only one vertex. In this paper, we determine the total domination numbers of lollipop graphs, umbrella graphs, and coconut graphs, and especially their $\gamma$-total dominating graphs.On the comaximal graph of a non-quasi-local atomic domain
https://comb-opt.azaruniv.ac.ir/article_14754.html
Let $R$ be an atomic domain such that $R$ has at least two maximal ideals. Let $Irr(R)$ denote the set of all irreducible elements of $R$ and let $J(R)$ denote the Jacobson radical of $R$. &nbsp;Let $\mathcal{I}(R) = \{R\pi\mid \pi\in Irr(R)\backslash J(R)\}$. In this paper, &nbsp;with $R$, we associate an undirected &nbsp;graph denoted by $\mathbb{CGI}(R)$ &nbsp;whose vertex set is $\mathcal{I}(R)$ and distinct vertices $R\pi_{1}$ and &nbsp;$R\pi_{2}$ are adjacent if and only if $R\pi_{1} + R\pi_{2} = R$. &nbsp;The aim of this paper is to study the interplay between some graph properties of $\mathbb{CGI}(R)$ and the algebraic properties of $R$.&nbsp;Sharp bounds on additively weighted Mostar index of Cacti
https://comb-opt.azaruniv.ac.ir/article_14759.html
Let C(n, t) denotes the collection of all cacti of order n with exactly t cycles and Ctndenotes the cacti of order n and t end vertices. In this paper, we compute the upper bound, second largest upper bound, and third largest upper bound of the additively weighted Mostar index of graphs in C(n, t). We also determine the upper bound of the additively weighted Mostar index for cacti of order n with a fixed number of end vertices. We characterize all the graphs attaining the bounds.Hypergraphs defined on algebraic structures
https://comb-opt.azaruniv.ac.ir/article_14760.html
There has been a great deal of research on graphs defined on algebraic structures in the last two decades. Power graphs, commuting graphs, cyclic graphs are some examples. In this paper we begin an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this can add a new perspective.A Classification of Graphs Through Quadratic Embedding Constants and Clique Graph Insights
https://comb-opt.azaruniv.ac.ir/article_14761.html
The quadratic embedding constant (QEC) of a graph $G$ is a new numeric invariant, which is defined in terms of the distance matrix and is denoted by $\mathrm{QEC}(G)$. By observing graph structure of the maximal cliques&nbsp;(clique graph), we show that a graph $G$ with $\mathrm{QEC}(G)&lt;-1/2$ admits a ``cactus-like'' structure. We derive a formula for the quadratic embedding constant of a graph consisting of two maximal cliques. As an application we discuss characterization of graphs along the increasing sequence of $\mathrm{QEC}(P_d)$, where $P_d$ is the path on $d$ vertices. In particular, we determine graphs $G$ satisfying $\mathrm{QEC}(G)&lt;\mathrm{QEC}(P_5)$.2-Rainbow Domination Number of the Subdivision of Graphs
https://comb-opt.azaruniv.ac.ir/article_14775.html
Let $G$ be a simple graph and $f : V (G) \rightarrow P(\{1,2\})$ be a function where for each vertex $v \in V (G)$ with $f(v)= \emptyset$ we have $\bigcup_{u \in N_{G}(v)} f(u) = \{1,2\}.$ Then $f$ is a $2$-rainbow dominating function (a $2RDF$) of $G.$ The &nbsp;weight of $f$ is $\omega(f)=\sum_{v \in V(G)} |f(v)|.$ The minimum weight among all of $2-$rainbow dominating functions is $2-$rainbow domination number &nbsp;and is denoted by $\gamma_{r2}(G)$. In this paper, &nbsp;we provide some bounds for the $2-$rainbow domination number of the subdivision graph $S(G)$ of &nbsp;a graph $G$. Also, among some other interesting results, we determine the exact value of $\gamma_{r2}(S(G))$ when $G$ is a tree, a bipartite graph, $K_{r,s}$, $K_{n_1,n_2,\dots,n_k}$ and $K_n$.Optimization problems with nonconvex multiobjective generalized Nash equilibrium problem constraints
https://comb-opt.azaruniv.ac.ir/article_14781.html
This work discusses a category of optimization problems in which the lower-level problems include multiobjective generalized Nash equilibrium problems. Despite the fact that it has various possible applications, there has been little research into it in the literature. We provide a single-level reformulation for these types of problems and highlight their equivalence in terms of global and local minimizers. Our method consists of transforming our problem into a one-level optimization problem, utilizing the kth-objective weighted-constraint and optimal value reformulation. The Mordukhovich generalized differentiation calculus is then used to derive completely detailed first-order necessary optimality conditions in the smooth setting.Strength based domination in graphs
https://comb-opt.azaruniv.ac.ir/article_14782.html
Let $G=(V,E)$ be a connected graph. Let $A\subseteq V$ and $v\in V-A.$ The dominating strength of $A$ on $v$ is defined by $s(v,A)=\sum\limits_{u\in A}\frac{1}{d(u,v)}.$ A subset $D$ of $V$ is called a strength based dominating set if for every vertex $v\notin D,$ there exists a subset $A$ of $D$ such that $s(v,A)\geq 1.$ The $sb$-domination number $\gamma_{sb}(G)$ is the minimum cardinality of a strength based dominating set of $G.$ In this paper we initiate a study of this parameter and indicate directions for further research.Two techniques to reduce the Pareto optimal solutions in multiobjective optimization problems
https://comb-opt.azaruniv.ac.ir/article_14783.html
In this study, for a decomposed multi-objective optimization problem, we &nbsp;propose the direct sum of the preference matrices of the &nbsp;subproblems provided by the decision maker (DM). Then, using this matrix, we present a new generalization of the rational efficiency concept for solving the multi-objective optimization problem (MOP). A problem that sometimes occurs in multi-objective optimization is the existence of a large set of Pareto optimal solutions. Hence, decision making based on selecting a unique preferred solution becomes difficult. Considering models with the concept of generalized rational efficiency can relieve some of the burden from the DM by shrinking the solution set. This paper discusses both theoretical and practical aspects of rationally efficient solutions related to this concept. Moreover, we present two techniques to reduce the Pareto optimal solutions using. The first technique involves using the powers of the preference matrix, while the second technique involves creating a new preference matrix by modifying the decomposition of the MOP.New bounds on distance Estrada index of graphs
https://comb-opt.azaruniv.ac.ir/article_14785.html
For a connected graph $G$ with vertex set $\{v_1,\ldots,v_n\}$, the distance matrix of $G$, denoted by $D(G)$, is an $n\times n$ matrix with zero main diagonal, such that its $(i,j)$-entry is $d(v_i,v_j)$, where $i\neq j$ and $d(v_i,v_j)$ is the distance between $v_i$ and $v_j$. Let $\theta_1,\ldots,\theta_n$ be the eigenvalues of $D(G)$. The distance Estrada index of $G$ is defined as $DEE(G)=\sum_{i=1}^ne^{\theta_i}$. In this paper we find some new sharp bounds for the distance Estrada index of graphs. Our results improve the previous bounds on the distance Estrada index of graphs.Edge Corona Product And Its Topological Descriptors with Applications in Complex Molecular Structures
https://comb-opt.azaruniv.ac.ir/article_14786.html
Graph operations offer a robust framework that enables the analysis, modeling, and resolution of intricate problems. Their versatility and broad range of applications make them essential across numerous fields of study and research, playing an irreplaceable role in tackling complex challenges. A topological index is a real number associated with a graph that gives insight into the topological properties of the graph. There are numerous topological indices in this era now, with three variants like degree based, distance based and eccentricity based topological indices. In this paper, we studied a well known graph operation named as edge corona product and investigate their some degree based topological indices. As applications, this graph operations can be used to study topological properties of complex structure of linear and cyclic silicate networks, together with triangular and double triangular networks. Some existing results in the literature can be obtained as corollaries of the new results. A conjecture is proposed relating the general first Zagreb index of the edge corona product of two graphs.A note on graphs with integer Sombor index
https://comb-opt.azaruniv.ac.ir/article_14789.html
For a graph $G$, the Sombor index of $G$ is defined as $ SO(G)=\sum_{uv\in E(G)} \sqrt{\deg(u)^2+\deg(v)^2}$, where $\deg(u)$ is referring to the degree of vertex $u$ in $G$. In this paper, we present a construction, namely $R_k$-construction which produce infinitely many families of graphs whose Sombor indices are integers.Remarks on the Bounds of Graph Energy
https://comb-opt.azaruniv.ac.ir/article_14795.html
Let $G$ be a graph of order $n$ with eigenvalues $\lambda _{1}\geq \lambda_{2}\geq \cdots \geq \lambda _{n}.$ The energy of $G$ is defined as $E\left(G\right) =\sum_{i=1}^{n}\left\vert \lambda _{i}\right\vert $. In the present paper, new bounds on $E(G)$ are provided. In addition, some bounds of $E(G)$ are compared.&nbsp;On the $A_{\alpha}$-spectrum of the $k$-splitting signed graph and neighbourhood coronas
https://comb-opt.azaruniv.ac.ir/article_14796.html
Let $\Sigma=(G,\sigma)$ be a signed graph with adjacency matrix $A(\Sigma)$ and $D(G)$ be the diagonal matrix of its vertex degrees. For any real $\alpha\in [0,1]$, the $A_{\alpha}$-matrix of a signed graph $\Sigma$ is defined as $A_{\alpha}(\Sigma)=\alpha D(G)+(1-\alpha)A(\Sigma)$. Given a signed graph $\Sigma$ with vertex set $V=\{v_1, v_2,\dots, v_n\}$, the $k$-splitting signed graph $SP_k(\Sigma)$ of $\Sigma$ is obtained by adding to each vertex $v\in V(\Sigma)$ new $k$ vertices say $u^1, u^2, \ldots, u^k$ and joining every neighbour say $u$ of the vertex $v$ to $u^i$, $1\le i\le k$ by an edge which inherits the sign from $uv$. In this paper, we determine the $A_{\alpha}$-spectrum of $SP_k(\Sigma)$ in case of $\Sigma$ being a regular signed graph. For $k=1$, we introduce two distinct coronas of signed graphs $\Sigma_1$ and $\Sigma_2$ based on $SP_1(\Sigma_1)$, namely the splitting V-vertex neighbourhood corona and the splitting S-vertex neighbourhood corona. By examining the $A_{\alpha}$-characteristic polynomial of the resulting signed graphs, we derive their $A_{\alpha}$-spectra under certain regularity conditions on the constituent signed graphs. As applications, we use these results to construct infinite pairs of nonregular $A_{\alpha}$-cospectral signed graphs.On the reciprocal distance Laplacian spectral radius of graphs
https://comb-opt.azaruniv.ac.ir/article_14803.html
The reciprocal distance Laplacian matrix of a connected graph $G$ is defined as $RD^L(G)=RTr(G)-RD(G)$, where $RTr(G)$ is the diagonal matrix whose $i$-th element $RTr(v_i)=\sum_{i\ne j\in V(G)} \frac{1}{d_{ij}}$ and $RD(G)$ is the Harary matrix. $RD^L(G)$ is a real symmetric matrix and we denote its eigenvalues as $\lambda_1(RD^L(G))\geq \lambda_2(RD^L(G))\geq\dots\geq\lambda_n(RD^L(G))$. The largest eigenvalue $\lambda_1(RD^L(G))$ of $RD^L(G)$ is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain upper bounds for the reciprocal distance Laplacian spectral radius. We characterize the extremal graphs attaining this bound.On the Energy of the line graph of Unitary Cayley graphs
https://comb-opt.azaruniv.ac.ir/article_14804.html
The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of the line graph of graph $G$ is denoted by $E(L(G))$.&nbsp; The unitary Cayley graph $X_n$ is a graph with the vertex set $Z_n=\{0, 1, \ldots, n-1\}$ and the edge set $\{(a,b) \, : \, ged(a-b,n)=1\}$. In this paper, we focus on the line graph of the unitary Cayley graph $X_n$ and compute the spectrum of line graphs of $X_n$ and its complement graph $\overline{X_n}$. We also obtain the energy of the line graph of $X_n$ and $\overline{X_n}$.On cozero divisor graphs of ring $Z_n$
https://comb-opt.azaruniv.ac.ir/article_14806.html
The cozero divisor graph $\Gamma^{\prime}(R)$ of a commutative ring $R$ &nbsp;is a simple graph with vertex set as non-zero zero divisor elements of $R$ such that two distinct vertices $x$ and $y$ are adjacent &nbsp;iff $x\notin Ry$ and $y\notin Rx$, where $xR$ is the ideal generated by $x$. In this article we find the spectra of $\Gamma^{\prime}(\mathbb{Z}_{n}) $ for $n\in \{q_{1}q_{2}, q_{1}q_{2}q_{3},q_{1}^{n_{1}}q_{2}\},$ where $q_{i}$'s are primes. As a consequence we obtain the bounds for the largest (smallest) eigenvalues, bounds for spread, rank and inertia of $ \Gamma^{\prime}(\mathbb{Z}_{q_{1}^{n_{1}}q_{2}})$ along with the determinant, inverse and square of trace of its quotient matrix. We present the extremal bounds for the energy of $\Gamma^{\prime}(\mathbb{Z}_{n})$ for $n=q_{1}^{n_{1}}q_{2}$ and characterize the extremal graphs attaining them. We close article with conclusion for furtherance.A traffic-based model to the $p$-median problem in congested networks
https://comb-opt.azaruniv.ac.ir/article_14807.html
In real urban transportation networks, the traffic counts of links affect their travel times, so considering fixed traffic-independent link travel times causes a lack of reliability in network models. In this paper, a traffic congestion model to the capacitated $p$-median problem is introduced to evaluate the effect of congestion on determining the optimal locations of facilities and allocating demands. Limited capacities for both nodes and links are considered, where increasing the traffic flow on one link would inevitably increase its travel time. Therefore, aside from considering the capacities of candidate points, the proposed model aims to locate facilities at nodes where their connected links also have enough capacities for passing through. Also, the effect of congestion imposed by the current flows corresponding to the existing origin-destination pairs in the network is considered. A generalized Benders decomposition algorithm is applied to reduce the problem to more manageable sub-problems, solved by the sub-gradient algorithm through consecutive iterations.Edge metric dimension of silicate networks
https://comb-opt.azaruniv.ac.ir/article_14808.html
Metric dimension is an essential parameter in graph theory that aids in addressing issues pertaining to information retrieval, localization, network design, and chemistry through the identification of the least possible number of elements necessary to identify the vertices in a graph uniquely. A variant of metric dimension, called the edge metric dimension focuses on distinguishing the edges in a graph $G$, with a vertex subset. The minimum possible number of vertices in such a set is denoted as $\dim_E(G)$. This paper presents the precise edge metric dimension of silicate networks.Some new families of KP-digraphs
https://comb-opt.azaruniv.ac.ir/article_14812.html
A kernel $N$ of a digraph $D$ is an independent set of vertices that is absorbent (for every vertex $u\in V(D)\setminus N$, there is a vertex $v\in N$ such that $(u,v)\in A(D)$). Let $D$ be a digraph such that every proper induced subdigraph has a kernel. If $D$ has a kernel, then $D$ is a \emph{kernel perfect digraph} (KP-digraph); otherwise, $D$ is a \emph{critical kernel imperfect digraph} (CKI-digraph). A digraph with the property $P$ is a digraph such that whenever a vertex reaches two other vertices through asymmetrical arcs, then these two vertices have the same out-neighborhood. In particular, digraphs whose asymmetrical part is a disjoint union of cycles have the property $P$. &nbsp; In this work, KP-digraphs with the property $P$ are characterized. As a consequence, KP-digraphs whose asymmetrical part is a Hamiltonian cycle are also characterized. For digraphs with a Hamiltonian cycle $\gamma$ as their asymmetrical part and whose diagonals are symmetrical of length 2, two algorithms are presented; &nbsp;the first one determines whether a digraph is a KP-digraph or a CKI-digraph, and the second constructs the kernel of the original digraph if it is a KP-digraph. As a consequence, a characterization of all CKI-digraphs whose asymmetrical part is a Hamiltonian cycle and whose diagonals are symmetrical of length 2 is shown.On subdivisions of oriented cycles in Hamiltonian digraphs with small chromatic number
https://comb-opt.azaruniv.ac.ir/article_14813.html
Cohen et al. conjectured that for each oriented cycle $C$, there is a smallest positive integer $f(C)$ such that every $f(C)$-chromatic strong digraph contains a subdivision of $C$. Let $C$ be an oriented cycle on $n$ vertices. For the class of Hamiltonian digraphs, El Joubbeh proved that $f(C)\leq 3n$. In this paper, we improve El Joubbeh's result by showing that $f(C)\leq 2n$ for the class of Hamiltonian digraphs.On the inverse problem of some bond additive indices
https://comb-opt.azaruniv.ac.ir/article_14814.html
Inverse Problem of topological indices deals with establishing whether or not a given number is a topological index of some graph. In this paper, we study the inverse topological index problem of some bond additive indices. In [1], it was conjectured that every positive integer except finitely many can be the Mostar index and edge Mostar index of some $c-$cyclic graph. We solve this conjecture for tricyclic graphs. We also study the inverse Albertson index problem and inverse sigma index problem for cacti and for cyclic graphs.Builder-Blocker general position games
https://comb-opt.azaruniv.ac.ir/article_14815.html
This paper considers a game version of the general position problem in which a general position set is built through adversarial play. Two players in a graph, Builder and Blocker, take it in turns to add a vertex to a set, such that the vertices of this set are always in general position. The goal of Builder is to create a large general position set, whilst the aim of Blocker is to frustrate Builder's plans by making the set as small as possible. The game finishes when no further vertices can be added without creating three-in-a-line and the number of vertices in this set is the game general position number. We determine this number for some common graph classes and provide sharp bounds, in particular for the case of trees. We also discuss the effect of changing the order of the players.Constant-Ratio Polynomial Time Approximation of the Asymmetric Minimum Weight Cycle Cover Problem with Limited Number of Cycles
https://comb-opt.azaruniv.ac.ir/article_14816.html
We consider polynomial time approximation of the minimum cost cycle cover problem for an edge-weighted digraph, where feasible covers are restricted to have at most $k$ disjoint cycles. In the literature this problem is referred to as Minimum-weight $k$-Size Cycle Cover Problem. &nbsp;The problem is closely related to the classic Traveling Salesman Problem and Vehicle Routing Problem and has many important applications in algorithms design and operations research. Unlike its unconstrained variant, the studied problem is strongly NP-hard even on undirected graphs and remains intractable in very specific settings. For any metric, the problem can be approximated in polynomial time within ratio 2, while &nbsp; in fixed-dimensional Euclidean spaces it admits polynomial time approximation schemes. In the same time, approximation of the more general asymmetric variant of the problem still remained weakly studied. In this paper, we propose the first constant-ratio approximation algorithm for this problem, which extends the recent breakthrough results of Svensson-Tarnawski-V&eacute;gh and Traub-Vygen for the Asymmetric Traveling Salesman Problem.Graceful Coloring of some Corona graphs - An algorithmic approach
https://comb-opt.azaruniv.ac.ir/article_14817.html
A graceful $k$-coloring of a non-empty graph G is a proper vertex coloring with $k$ colors that induces a proper edge coloring, where the color for an edge $uv$ is the absolute difference between the colors assigned to the vertices $u$ and $v$. The minimum $k$ for which $G$ admits a graceful $k$-coloring is called the graceful chromatic number of $G$ ($\chi_ {g} (G)$). The problem of determining the graceful chromatic number for some corona graphs with the related coloring algorithms are studied in this paper.Independent transversal domination subdivision number of trees
https://comb-opt.azaruniv.ac.ir/article_14822.html
A set $S\subseteq V $ of vertices in a graph $G=(V,E)$ is called a dominating set if every vertex in $V \setminus S $ is adjacent to a vertex in S. The domination number $\gamma(G)$ is the minimum cardinality of a dominating set of $G$. The domination subdivision number $sd_{\gamma}(G)$ is the minimum number of edges that must be subdivided (each edge in $G$ can be subdivided at most once) in order to increase the domination number. Sahul Hamid &nbsp;defined a dominating set which intersects every maximum independent set in $G$ to be an \textit{independent transversal dominating set. The minimum cardinality of an independent transversal dominating set is called the independent transversal domination number} of $G$ and is denoted by $\gamma_{it}(G)$. We extend the idea of domination subdivision number &nbsp;to independent transversal domination. &nbsp;The independent transversal domination subdivision number of a graph $G$ denoted by $sd_{\gamma_{it}}(G)$ is the minimum number of edges that must be subdivided (each edge in $G$ can be subdivided at most once) in order to increase the independent transversal domination number. In this paper we initiate a study of this parameter with respect to trees.Irredundance chromatic number and gamma chromatic number of trees
https://comb-opt.azaruniv.ac.ir/article_14825.html
A vertex subset $S$ of a graph $G = (V, E)$ is irredundant if every vertex in $S$ has a private neighbor with respect to $S$. An irredundant set $S$ of $G$ is maximal if, for any $v \in V - S$, the set $S \cup \{v\}$ is no longer irredundant. The lower irredundance number of $G$ is the minimum cardinality of a maximal irredundant set of $G$ and is denoted by $ir(G)$. A coloring $\mathcal{C}$ of $G$ is said to be the irredundance coloring if there exists a maximal irredundant set $R$ of $G$ such that all the vertices of $R$ receive different colors. The minimum number of colors required for an irredundance coloring of $G$ is called the irredundance chromatic number of $G$, and is denoted by $\chi_{i}(G)$. A coloring $\mathcal{C}$ of $G$ is said to be the gamma coloring if there exists a dominating set $D$ of $G$ such that all the vertices of $D$ receive different colors. The minimum number of colors required for a gamma coloring of $G$ is called the gamma chromatic number of $G$, and is denoted by $\chi_{\gamma}(G)$. In this paper, we prove that every tree $T$ satisfies $\chi_{i}(T) = ir(T)$ unless $T$ is a star. Further, we prove that $\gamma(T) \leq \chi_{\gamma}(T) \leq \gamma(T) + 1$. We characterize all trees satisfying the upper bound.Additive closedness in subsets of $\mathbb{Z}_n$
https://comb-opt.azaruniv.ac.ir/article_14827.html
The r-value in subsets of finite abelian groups serves as a metric for evaluating the degree of closedness within these subsets. The notion of the r-value is intricately linked to other mathematical constructs such as sum-free sets, Sidon sets, and Schur triples. We extend the definition of r-value of a subset in a finite abelian group and investigate the r-values of subsets of Z_n, by constructing a formula for r-values of intervals consist of consecutive residue classes modulo n.Weak signed Roman $k$-domatic number of a digraph
https://comb-opt.azaruniv.ac.ir/article_14828.html
Let $D$ be a digraph with vertex set $V(D)$, and let $k\ge 1$ be an integer. A weak signed Roman $k$-dominating function on a digraph $D$ is a function &nbsp;$f:V (D)\longrightarrow \{-1, 1, 2\}$ such that $\sum_{u\in N^-[v]}f(u)\ge k$ for every $v\in V(D)$, where $N^-[v]$ consists of $v$ and all vertices of $D$ from which arcs go into $v$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct weak signed Roman $k$-dominating functions on $D$ with the property that $\sum_{i=1}^df_i(v)\le k$ for each $v\in V(D)$, is called a &nbsp;weak signed Roman $k$-dominating family (of functions) on $D$. The maximum number of functions in a &nbsp;weak signed Roman $k$-dominating family on $D$ is the &nbsp;weak signed Roman $k$-domatic number of $D$, denoted by $d_{wsR}^k(D)$. In this paper we initiate the study of the weak signed Roman $k$-domatic number in digraphs, and we present sharp bounds for $d_{wsR}^k(D)$. In addition, we determine the weak signed Roman $k$-domatic number of some digraphs.Line graph characterization of the order supergraph of a finite group
https://comb-opt.azaruniv.ac.ir/article_14829.html
The power graph ${\mathcal{P}(G)}$ is the simple undirected graph with group elements as a vertex set and two elements are adjacent if one of them is a power of the other. The order supergraph ${\mathcal{S}(G)}$ of the power graph ${\mathcal{P}(G)}$ is the simple undirected graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if $o(x)\vert o(y)$ or $o(y)\vert o(x)$. In this paper, we classify all the finite groups $G$ such that the order supergraph ${\mathcal{S}(G)}$ is the line graph of some graph. Moreover, we characterize finite groups whose order supergraphs are the complement of line graphs.Mixed double Roman domination in graphs
https://comb-opt.azaruniv.ac.ir/article_14830.html
Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. A mixed double Roman dominating function (MDRDF) of $G$ is a function $f:V\cup E \rightarrow \{0,1,2,3\}$ satisfying &nbsp;(1) for any element $x\in V\cup E$ with $f(x)=0$, there must be either element $y\in V\cup E$, with $f(y)=3,$ &nbsp;which is adjacent or incident to $x$, or either two elements $y,z \in V\cup E$, with $f(y),f(z)=2$ which are adjacent or incident to $x$; (2) for any element $x\in V\cup E$ with $f(x)=1$, there must be either &nbsp;element $y\in V\cup E$, with $f(y)\ge 2,$ &nbsp;which is adjacent or incident to $x$. The weight of an MDRDF $f$ is $w(f)=f(V\cup E)=\sum_{x\in V\cup E} f(x)$ and the minimum weight among all the MDRD functions the \textit{MDRD-number}, $\gamma _{dR}^*(G)$, of the graph $G$. In this paper we start the study of this variation of the classic Roman domination problem by setting some basic results, giving exact values and sharp bounds of the MDRD number and we approach the study of the complexity of the decision problem associated to the MDR domination in graphs.&nbsp;Hyperbolic $k$-Mersenne and $k$-Mersenne-Lucas Quaternions with it’s associated Spinor algebra
https://comb-opt.azaruniv.ac.ir/article_14833.html
In this article, we introduce and study hyperbolic $k$-Mersenne and $k$-Mersenne-Lucas spinors. First, we give hyperbolic $k$-Mersenne and $k$-Mersenne-Lucas quaternions with some algebraic properties. Next we introduce the spinor family of $k$-Mersenne and $k$-Mersenne-Lucas numbers using the hyperbolic $k$-Mersenne and $k$-Mersenne-Lucas quaternions. Here, we start with Binet-type formulas and algebraic properties such as Catalan's identity, Cassini's identity, d'Ocagne's identity, etc. Additionally, we obtain various types of generating functions. Moreover, we give partial sum formulas in closed form.A complete characterization of spectra of the Randić matrix of level-wise regular trees
https://comb-opt.azaruniv.ac.ir/article_14834.html
Let $G$ be a simple finite connected graph with vertex set $V(G) = \{v_1,v_2,\ldots,v_n\}$ and $d_i$ be the degree of the vertex $v_i$. The Randić matrix $\R(G) = [r_{i,j}]$ of graph $G$ is an $n \times n$ matrix whose $(i,j)$-entry $r_{i,j}$ is $r_{i,j} = 1/\sqrt{d_id_j}$ if $v_i$ and $v_j$ are adjacent in $G$ and 0 otherwise. A level-wise regular tree is a tree rooted at one vertex $r$ or two (adjacent) vertices $r$ and $r'$ in which all vertices with the minimum distance $i$ from $r$ or $r'$ have the same degree $m_i$ for $0 \leq i \leq h$, where $h$ is the height of $T$. In this paper, we give a complete characterization of the eigenvalues with their multiplicity of the Randić matrix of level-wise regular trees. We prove that the eigenvalues of the Randić matrix of a level-wise regular tree are the eigenvalues of the particular tridiagonal matrices, which are formed using the degree sequence $(m_0,m_1,\ldots,m_{h-1})$ of level-wise regular trees.Neighborhood First Zagreb Index and Maximal Unicyclic and Bicyclic Graphs
https://comb-opt.azaruniv.ac.ir/article_14835.html
The Neighborhood First Zagreb Index $NM_{1}$ measures the topological properties of a molecular graph. Neighborhood First Zagreb Index $NM_{1}$ is defined as $NM_{1}(G) = \sum_{ v\in V (G)}(S(v))^{2}$, where $S(v)$ used to represent the sum of degrees of vertices adjacent to a vertex $v$ in a graph $G$. &nbsp;In this study, we focus on characterizing the graphs with the maximum neighborhood first Zagreb index in the class of unicyclic/bicyclic graphs on $n$ vertices, where $n$ is a fixed integer greater than or equal to $5$. Specifically, we are interested in identifying the graphs that have the highest value according to the recently introduced neighborhood first Zagreb index $NM_{1}$.Algorithmic Results on Independent Roman {2}-Domination
https://comb-opt.azaruniv.ac.ir/article_14836.html
An independent Roman $\{2\}$-dominating function (IR2DF) $f:V \rightarrow \{0, 1, 2\}$ in a graph $G=(V,E)$ has the properties that $\sum_{u\in N(v)}f(u) \geq 2$ if $f(v)=0$, and $f(u)=0$ for $u \in N(v)$ if $f(v) \geq 1$, where $v \in V$. The weight of an IR2DF in a graph $G$ is defined as the sum of its function values for all vertices, given by $\omega(f)=\sum_{v\in V}f(v)$. The independent Roman $\{2\}$-domination number of $G$, denoted $i_{\{R2\}}(G)$, is the minimum weight of all IR2DFs in $G$. In this paper, we prove that the independent Roman $\{2\}$-domination problem (IR2D) is NP-complete, even when restricted to chordal bipartite graphs. We then give an exact formula for the IR2D in corona graphs. Finally, we present two linear-time algorithms for solving IR2D for proper interval graphs and block-cactus graphs, respectively.On the Metric Dimension and Spectrum of Graphs
https://comb-opt.azaruniv.ac.ir/article_14837.html
The algebraic approach to graph theoretical problems has been extensively studied by looking at the spectrum of a graph's representation matrix. In this paper, we investigate some relationships between the metric dimension of a graph $G$ and its nullity, that is, the multiplicity of eigenvalue $0$ in the adjacency matrix of $G$, and the eigenvalues of its Laplacian and distance matrices. Furthermore, we also present a relationship between the metric dimension of a graph and its nullity, using twin classes.On relations between the modified hyper Wiener index and some degree based indices of trees
https://comb-opt.azaruniv.ac.ir/article_14838.html
Let T be a tree of order n with Laplacian eigenvalues $\mu_{1}\geq \mu_{2}\geq \cdots \geq \mu_{n-1}&gt;\mu_{n}=0$. The Wiener index of T is defined as $W(T)=n\sum_{i=1}^{n-1} \frac{1}{\mu_i }$. The modified hyper Wiener index of T is stated in terms of W(T) and Laplacian eigenvalues as $WWW(T)= \frac{W(T)^2}{2n}-\frac{n}{2}\sum_{i=1}^{n-1} \frac{1}{\mu_i^2}$. In this study, we present some relations between modified hyper-Wiener index, the first Zagreb index, modified first Zagreb index and inverse degree index of trees when order n and maximal vertex degree of a graph are known.Combinations without specified separations
https://comb-opt.azaruniv.ac.ir/article_14839.html
We consider the restricted subsets of $\mathbb{N}_n=\{1,2,\ldots,n\}$ with $q\geq1$ being the largest member of the set $\mathcal{Q}$ of disallowed differences between subset elements. We obtain new results on various classes of problem involving such combinations lacking specified separations. &nbsp;In particular, we find recursion relations for the number of $k$-subsets for any $\mathcal{Q}$ when $|{\mathbb{N}_q-\mathcal{Q}}|\leq2$. &nbsp;The results are obtained, in a quick and intuitive manner, as a consequence of a bijection we give between such subsets and the restricted-overlap tilings of an $(n+q)$-board (a linear array of $n+q$ square cells of unit width) with squares ($1\times1$ tiles) and combs. &nbsp;A $(w_1,g_1,w_2,g_2,\ldots,g_{t-1},w_t)$-comb is composed of $t$ sub-tiles known as teeth. &nbsp;The $i$-th tooth in the comb has width $w_i$ and is separated from the $(i+1)$-th tooth by a gap of width $g_i$. Here we only consider combs with $w_i,g_i\in\mathbb{Z}^+$. &nbsp;When performing a restricted-overlap tiling of a board with such combs and squares, the leftmost cell of a tile must be placed in an empty cell whereas the remaining cells in the tile are permitted to overlap other non-leftmost filled cells of tiles already on the board.Disproof of two conjectures on proper 2-dominating sets in graphs
https://comb-opt.azaruniv.ac.ir/article_14840.html
In this note, we disprove two conjectures recently stated on proper $2$-dominating sets in graphs. We recall that a proper $2$-dominating set of a graph $G=(V,E)$ is a subset $D$ of $V$ such that every vertex in $V-D$ has at least two neighbors in $D$ except for at least one vertex which must have exactly two neighbors in $D$.Seidel energy of a graph with self-loops
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 $S\subseteq V$ &nbsp;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 $v_i\not\in S$. &nbsp; &nbsp; 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.On Connected Graphs with Integer-Valued Q-Spectral Radius
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 &nbsp;$Q$-eigenvalue does not belong to the open interval $(0,1)$ and has an integral $Q$-spectral radius.The monophonic pebbling number of neural networks with generalized algorithm and their applications
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), &nbsp;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 $\sigma$.On the nullity of cycle-spliced T-gain graphs
https://comb-opt.azaruniv.ac.ir/article_14849.html
Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain (or complex unit gain) graph and $A(\Phi)$ be its adjacency matrix. The nullity of $\Phi$, denoted by $\eta(\Phi)$, is the multiplicity of zero as an eigenvalue of $A(\Phi)$, and the cyclomatic number of $\Phi$ is defined by $c(\Phi)=e(\Phi)-n(\Phi)+\kappa(\Phi)$, where $n(\Phi)$, $e(\Phi)$ and $\kappa(\Phi)$ are the number of vertices, edges and connected components of $\Phi$, respectively. A connected graph is said to be cycle-spliced if every block in it is a cycle. We consider the nullity of cycle-spliced $\mathbb{T}$-gain graphs. Given a cycle-spliced $\mathbb{T}$-gain graph $\Phi$ with $c(\Phi)$ cycles, we prove that $0 \leq \eta(\Phi)\leq c(\Phi)+1$. Moreover, we show that there is no cycle-spliced &nbsp;$\mathbb{T}$-gain graph $\Phi$ of any order with $\eta(\Phi)=c(\Phi)$ whenever there are no odd cycles whose gain has real part $0$. We give examples of cycle-spliced &nbsp;$\mathbb{T}$-gain graphs whose nullity equals the cyclomatic number, and we show some properties of those graphs $\Phi$ such that $\eta(\Phi)=c(\Phi)-\varepsilon$, $\varepsilon \in \{0,1\}$. A characterization is given in case $\eta(\Phi)=c(\Phi)$ when $\Phi$ is obtained by identifying a unique common vertex of $2$ cycle-spliced $\mathbb{T}$-gain graphs $\Phi_1$ and $\Phi_2$. Finally, we compute the nullity of all $\mathbb{T}$-gain graphs $\Phi$ with $c(\Phi)=2$.On the rainbow connection number of the connected inverse graph of a finite group
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.Unbalanced complete bipartite signed graphs ${K_{m, n}}^{\sigma}$ having $m$ and $n$ as Laplacian eigenvalues with maximum multiplicities
https://comb-opt.azaruniv.ac.ir/article_14855.html
A signed graph $ {G}^{\sigma} = (G, \sigma) $ consists of an underlying graph $ G = (V, E) $ along with a signature function $ \sigma: E \rightarrow \{-1, 1\} $. A cycle in a signed graph is termed positive if it contains an even number of negative edges, and negative if it contains an odd number of negative edges. A signed graph is considered { balanced} if it has no negative cycles; otherwise, it is { unbalanced}. Let $K_{m,n}$ be a { complete bipartite graph} on $m+n$ vertices. It is well known that for a balanced complete bipartite signed graph $ {K_{m,n}}^{\sigma} $, the parameters $ m $ and $ n $ are Laplacian eigenvalues with multiplicities $ n-1 $ and $ m-1 $, respectively. This raises a natural question about the maximum multiplicities of Laplacian eigenvalues $ m $ and $ n $ in an unbalanced complete bipartite signed graph $ {K_{m,n}}^{\sigma} $. In this paper, we demonstrate that the multiplicities of the Laplacian eigenvalues $ m $ and $ n $ in an unbalanced complete bipartite signed graph $ {K_{m,n}}^{\sigma} $ are at most $ n-2 $ and $ m-2 $, respectively. Additionally, we characterize all the signed graphs for which $ m $ and $ n $ are Laplacian eigenvalues with these maximum multiplicities.Weighted topological indices of graphs
https://comb-opt.azaruniv.ac.ir/article_14857.html
The definition of the weighted topological index associated with a degree function $\phi$ is $\Phi(G)=\sum_{uv\in E(G)}\phi(d_{u},d_{v})$, where $d_{u}$ denotes the degree of node $u$ and $\phi$ satisfies symmetric property $\phi(d_{u},d_{v})=\phi(d_{v},d_{u})$. In this paper, we characterized extremal graphs and presented several results concerning the function $\Phi(G)$ in terms of various graph invariants. Additionally, we characterize the graphs that achieve these bounds and present multiple bounds for $\Phi(G)$ for the class of cozero divisor graphs defined on commutative rings.Decomposition of complete graphs into disconnected bipartite graphs with seven edges and eight vertices
https://comb-opt.azaruniv.ac.ir/article_14858.html
In this paper, we continue investigation of decompositions of complete graphs into graphs with seven edges. The spectrum has been completely determined for such graphs with at most six vertices. Connected graphs with seven edges and seven vertices are necessarily unicyclic and the spectrum for bipartite ones was completely determined by the authors. Connected graphs with seven edges and eight vertices are trees and the spectrum was found by Huang and Rosa. As a next step in the quest of completing the spectrum for all graphs with seven edges, we completely solve the case of disconnected bipartite graphs with seven edges and eight vertices.