Communications in Combinatorics and Optimization
http://comb-opt.azaruniv.ac.ir/
Communications in Combinatorics and Optimizationendaily1Sat, 01 Jun 2024 00:00:00 +0430Sat, 01 Jun 2024 00:00:00 +0430Independence Number and Connectivity of Maximal Connected Domination Vertex Critical Graphs
http://comb-opt.azaruniv.ac.ir/article_14648.html
A $k$-CEC graph is a graph $G$ which has connected domination number $\gamma_{c}(G) = k$ and $\gamma_{c}(G + uv) &lt; k$ for every $uv \in E(\overline{G})$. A $k$-CVC graph $G$ is a $2$-connected graph with &nbsp;$\gamma_{c}(G) = k$ and $\gamma_{c}(G - v) &lt; k$ for any $v \in V(G)$. A graph is said to be maximal $k$-CVC if it is both $k$-CEC and $k$-CVC. Let $\delta$, $\kappa$, and $\alpha$ be the minimum degree, connectivity, and independence number of $G$, respectively. In this work, we prove that for a maximal $3$-CVC graph, if $\alpha = \kappa$, then $\kappa = \delta$. We additionally consider the class of maximal $3$-CVC graphs with $\alpha &lt; \kappa$ and $\kappa &lt; \delta$, and prove that every $3$-connected maximal $3$-CVC graph when $\kappa &lt; \delta$ is Hamiltonian connected.Reconfiguring Minimum Independent Dominating Sets in Graphs
http://comb-opt.azaruniv.ac.ir/article_14682.html
The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$, denoted $\mathscr{I}(G)$, is the graph whose vertices correspond to the $i(G)$-sets, and where two $i(G)$-sets are adjacent if and only if they differ by two adjacent vertices. We show that not all graphs are $i$-graph realizable, that is, given a target graph $H$, there does not necessarily exist a seed graph $G$ such that $H \cong \mathscr{I}(G)$. &nbsp;Examples of such graphs include $K_{4}-e$ and $K_{2,3}$. We build a series of tools to show that known $i$-graphs can be used to construct new $i$-graphs and apply these results to build other classes of $i$-graphs, such as block graphs, hypercubes, forests, cacti, and unicyclic graphs.A Counterexample on the Conjecture and bounds on χgd-number of Mycielskian of a graph
http://comb-opt.azaruniv.ac.ir/article_14512.html
A coloring $C=(V_1, \dots, V_k)$ of $G$ partitions the vertex set $V(G)$ into independent sets $V_i$ which are said to be color classes with respect to the coloring $C$. A vertex $v$ is said to have a dominator (dom) color class in $C$ if there is color class $V_i$ such that $v$ is adjacent to all the vertices of $V_i$ and $v$ is said to have an anti-dominator (anti-dom) color class in $C$ if there is color class $V_j$ such that $v$ is not adjacent to any vertex of $V_j$. Dominator coloring of $G$ is a coloring $C$ of $G$ such that every vertex has a dom color class. The minimum number of colors required for a dominator coloring of $G$ is called the dominator chromatic number of $G$, denoted by $\chi_{d}(G)$. Global Dominator coloring of $G$ is a coloring $C$ of $G$ such that every vertex has a dom color class and an anti-dom color class. The minimum number of colors required for a global dominator coloring of $G$ is called the global dominator chromatic number of $G$, denoted by $\chi_{gd}(G)$. In this paper, we give a counterexample for the conjecture posed in [I. Sahul Hamid, M.Rajeswari, Global dominator coloring of graphs, Discuss. Math. Graph Theory 39 &nbsp;(2019), 325--339] that for a graph $G$, if $\chi_{gd}(G)=2\chi_{d}(G)$, then $G$ is a complete multipartite graph. We deduce upper and lower bound for the global dominator chromatic number of Mycielskian of the graph $G$ in terms of dominator chromatic number of $G$.Graphoidally Independent Infinite Cactus
http://comb-opt.azaruniv.ac.ir/article_14473.html
A graphoidal cover of a graph $G$ (not necessarily finite) is a collection $\psi$ of paths (not necessarily finite, not necessarily open) satisfying the following axioms: (GC-1) Every vertex of $G$ is an internal vertex of at most one path in $\psi$, and (GC-2) every edge of $G$ is in exactly one path in $\psi$. The pair $(G, \psi)$ is called a graphoidally covered graph and the paths in $\psi$ are called the $\psi$-edges of $G$. In a graphoidally covered graph $(G, \psi)$, two distinct vertices $u$ and $v$ are $\psi$-adjacent if they are the ends of an open $\psi$-edge. A graphoidally covered graph $(G, \psi)$ in which no two distinct vertices are $\psi$-adjacent is called $\psi$-independent and the graphoidal cover $\psi$ is called a totally disconnecting graphoidal cover of $G$. Further, a graph possessing a totally disconnecting graphoidal cover is called a graphoidally independent graph. The aim of this paper is to establish complete characterization of graphoidally independent infinite cactus.Leech Graphs
http://comb-opt.azaruniv.ac.ir/article_14452.html
Let $t_p(G)$ denote the number of paths in a graph $G$ and let $f:E\rightarrow \mathbb{Z}^+$ be an edge labeling of $G$. The weight of a path $P$ is the sum of the labels assigned to the edges of $P$. If the set of weights of the paths in $G$ is $\{1,2,3,\dots,t_p(G)\}$, then $f$ is called a Leech labeling of $G$ and a graph which admits a Leech labeling is called a Leech graph. In this paper, we prove that the complete bipartite graphs $K_{2,n}$ and $K_{3,n}$ are not Leech graphs and determine the maximum possible value that can be given to an edge in the Leech labeling of a cycle.PI Index of Bicyclic Graphs
http://comb-opt.azaruniv.ac.ir/article_14514.html
The PI index of a graph $G$ is given by $PI(G)=\sum_{e\in E(G)}(\left|V(G)\right|-N_G(e))$, where $N_G(e)$ is &nbsp;the number of equidistant vertices for the edge $e$. Various topological indices of bicyclic graphs have already been calculated. In this paper, we obtained the exact value of the PI index of bicyclic graphs. We also explore the extremal graphs among all bicyclic graphs with respect to the PI index. Furthermore, we calculate the PI index of a cactus graph and determine the extremal values of the PI index among cactus graphs.Algorithmic complexity of triple Roman dominating functions on graphs
http://comb-opt.azaruniv.ac.ir/article_14504.html
Given a graph $G=(V,E)$, &nbsp;a &nbsp;function &nbsp;$f:V\to \{0,1,2,3,4\}$ is a triple Roman &nbsp;dominating function (TRDF) &nbsp;of $G$, for each vertex $v\in V$, &nbsp;(i) if $f (v ) = 0 $, then &nbsp;$v$ must have either one neighbour in $V_4$, or either two neighbours in $V_2 \cup &nbsp;V_3$ (one neighbour in $V_3$) or either three neighbours in $V_2 $, (ii) if $f (v ) = 1 $, then $v$ must have either one neighbour in &nbsp;$V_3 \cup &nbsp;V_4$ &nbsp;or either two neighbours in $V_2 $, and if $f (v ) = 2 $, then $v$ must have one neighbour in $V_2 \cup &nbsp;V_3\cup &nbsp;V_4$. The triple Roman &nbsp;domination number of $G$ is the &nbsp;minimum weight of an TRDF &nbsp;$f$ &nbsp;of $G$, where the weight of $f$ is $\sum_{v\in V}f(v)$. &nbsp;The triple &nbsp;Roman &nbsp;domination problem is to compute the &nbsp;triple Roman &nbsp;domination number of a given graph. &nbsp;In this paper, we study the triple &nbsp;Roman &nbsp;domination problem. We show that &nbsp; the problem is NP-complete for &nbsp;the &nbsp;star convex bipartite &nbsp;and the &nbsp; comb convex bipartite graphs and is APX-complete for graphs of degree at~most~4. We propose a linear-time algorithm for computing &nbsp;the triple Roman &nbsp;domination number of proper interval graphs. &nbsp;We also &nbsp; give an $( 2 H(\Delta(G)+1) -1 &nbsp;)$-approximation algorithm &nbsp;for solving the problem &nbsp;for any graph $G$, &nbsp;where &nbsp; $ &nbsp;\Delta(G)$ is the maximum degree of $G$ and $H(d)$ denotes the first $d$ terms of the harmonic &nbsp;series. In addition, we prove &nbsp;that &nbsp;for any $\varepsilon&gt;0$ &nbsp;there is no &nbsp;$(1/4-\varepsilon)\ln|V|$-approximation &nbsp;polynomial-time &nbsp; algorithm for solving &nbsp;the problem on bipartite and split &nbsp;graphs, unless NP $\subseteq$ DTIME $(|V|^{O(\log\log|V |)})$.Coalition of cubic graphs of order at most $10$
http://comb-opt.azaruniv.ac.ir/article_14542.html
The coalition in a graph $G$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a dominating set but whose union $V_{1}\cup &nbsp;V_{2}$, is a dominating set. A coalition partition in a graph $G$ is a vertex partition $\pi$ = $\{V_1, V_2,\dots, V_k \}$ such that every set $V_i \in \pi$ is not a dominating set but forms a coalition with another set $V_j\in \pi$ which is not a dominating set. The coalition number $C(G)$ equals the maximum $k$ of a coalition partition &nbsp;of $G$. In this paper, we compute the coalition numbers of all cubic graphs of order at most $10$.Total Chromatic Number for Certain Classes of Lexicographic Product Graphs
http://comb-opt.azaruniv.ac.ir/article_14478.html
A total coloring of a graph $G$ is an assignment of colors to all the elements (vertices and edges) of the graph in such a way that no two adjacent or incident elements receive the same color. The total chromatic number of $G$, denoted by $\chi''(G)$, is the minimum number of colors which need for total coloring of $G$. The Total Coloring Conjecture (TCC) made independently by Behzad and Vizing which claims that, $\Delta(G)+1 \leq \chi''(G) \leq \Delta(G)+2 $, where $\Delta(G)$ is the maximum degree of $G$. The lower bound is sharp and the upper bound remains to be proved. In this paper, we prove the TCC for certain classes of lexicographic and deleted lexicographic products of graphs. Also, we obtained the lower bound for certain classes of these products.On the rna number of generalized Petersen graphs
http://comb-opt.azaruniv.ac.ir/article_14518.html
A signed graph $(G,\sigma)$ is called a parity signed graph if there exists a bijective mapping $f \colon V(G) \rightarrow \{1,\ldots,|V(G)|\}$ such that for each edge $uv$ in $G$, $f(u)$ and $f(v)$ have same parity if $\sigma(uv)=+1$, and opposite parity if $\sigma(uv)=-1$. The \emph{rna} number $\sigma^{-}(G)$ of $G$ is the least number of negative edges among all possible parity signed graphs over $G$. Equivalently, $\sigma^{-}(G)$ is the least size of an edge-cut of $G$ that has nearly equal sides.In this paper, we show that for the generalized Petersen graph $P_{n,k}$, $\sigma^{-}(P_{n,k})$ lies between $3$ and $n$. Moreover, we determine the exact value of $\sigma^{-}(P_{n,k})$ for $k\in \{1,2\}$. The \emph{rna} numbers of some famous generalized Petersen graphs, namely, Petersen graph, D\" urer graph, M\" obius-Kantor graph, Dodecahedron, Desargues graph and Nauru graph are also computed. Recently, Acharya, Kureethara and Zaslavsky characterized the structure of those graphs whose \emph{rna} number is $1$. We use this characterization to show that the smallest order of a $(4n+1)$-regular graph having \emph{rna} number $1$ is $8n+6$. We also prove the smallest order of $(4n-1)$-regular graphs having \emph{rna} number $1$ is bounded above by $12n-2$. In particular, we show that the smallest order of a cubic graph having \emph{rna} number $1$ is 10.The crossing numbers of join product of four graphs on six vertices with discrete graphs
http://comb-opt.azaruniv.ac.ir/article_14527.html
The main aim of the paper is to give the crossing number of the join product $G^\ast + D_n$ for the graph $G^\ast$ isomorphic to 4-regular graph on six vertices except for two distinct edges with no common vertex such that two remaining vertices are still adjacent, and where $D_n$ consists of $n$ isolated vertices. The proofs are done with possibility of an existence of a separating cycle in some particular drawing of the investigated graph $G^\ast$ and also with the help of well-known exact values for crossing numbers of join products of two subgraphs $H_k$ of $G^\ast$ with discrete graphs.Optimal Coverage of Borders Using Unmanned Aerial Vehicles
http://comb-opt.azaruniv.ac.ir/article_14528.html
Unmanned Aerial Vehicles (UAVs) play a very important role in military and civilian activities. In this paper, the aim is to cover the borders of Iran using UAVs. For this purpose, two zero-one programming &nbsp;models are presented. In the first model, our goal is to cover the borders of Iran at the minimum total time (the required time to prepare UAVs to start flying and the flight time of the UAVs). In this model, by minimizing the total time of UAVs for covering the borders, the costs appropriate to the flight of UAVs (such as the fuel costs of UAVs) are also reduced. In the second model, which is mostly used in emergencies and when a military attack occurs on the country's borders, the aim is to minimize the maximum required time to counter attacks and cover the entire country's borders. The efficiency of both models is shown by numerical examples.Balance theory: An extension to conjugate skew gain graphs
http://comb-opt.azaruniv.ac.ir/article_14466.html
We extend the notion of balance from the realm of signed and gain graphs to conjugate skew gain graphs which are skew gain graphs where the labels on the oriented edges get conjugated when we reverse the orientation. We characterize the balance in a conjugate skew gain graph in several ways especially by dealing with its adjacency matrix and the $g$-Laplacian matrix. We also deal with the concept of anti-balance in conjugate skew gain graphs.New results on Orthogonal Component Graphs of Vector Spaces over $\mathbb{Z}_p$
http://comb-opt.azaruniv.ac.ir/article_14532.html
A new concept known as the orthogonal component graph associated with a finite-dimensional vector space over a finite field has been recently added as another class of algebraic graphs. In these graphs, the vertices will be all the possible non-zero linear combinations of orthogonal basis vectors. Any two vertices will be adjacent if the corresponding vectors are orthogonal. In this paper, we discuss the various colorings and structural properties of orthogonal component graphs.The Length of the Longest Sequence of Consecutive FS-double Squares in a Word
http://comb-opt.azaruniv.ac.ir/article_14492.html
A square is a concatenation of two identical words, and a word $w$ is said to have a square $yy$ if $w$ can be written as $xyyz$ for some words $x$ and $z$. It is known that the ratio of the number of distinct squares in a word to its length is less than two, and any location of a word could begin with two distinct squares which are appearing in the word for the last time. A square whose first location starts with the last occurrence of two distinct squares is an FS-double square. We explore and identify the conditions under which a sequence of locations in a word starts with FS-double squares. We first find the structure of a word that begins with two consecutive FS-double squares and obtain its properties that enable us to extend the sequence of FS-double squares. It is proved that the length of the longest sequence of consecutive FS-double squares in a word of length $n$ is at most $\frac{n}{7}$. We show that the squares in the longest sequence of consecutive FS-double squares are conjugates.On the anti-forcing number of graph powers
http://comb-opt.azaruniv.ac.ir/article_14549.html
Let $G=(V,E)$ be a simple connected graph. A perfect matching (or Kekul\'e structure in chemical literature) of $G$ is a set of disjoint edges which covers all vertices of $G$. The anti-forcing number of $G$ is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching and is denoted by $af(G)$. &nbsp; &nbsp; For every $m\in\mathbb{N}$, the $m$th power of $G$, denoted by $G^m$, is a graph with the same vertex set as $G$ such that two vertices are adjacent in $G^m$ if and only if their distance is at most $m$ in $G$. In this paper, we study the anti-forcing number of the powers of some graphs.Mathematical results on harmonic polynomials
http://comb-opt.azaruniv.ac.ir/article_14660.html
The harmonic polynomial was defined in order to understand better the harmonic topological index. Here, we obtain several properties of this polynomial, and we prove that several properties of a graph can be deduced from its harmonic polynomial. Also, we prove that graphs with the same harmonic polynomial share many properties although they are not necessarily isomorphic.On the vertex irregular reflexive labeling of generalized friendship graph and corona product of graphs
http://comb-opt.azaruniv.ac.ir/article_14545.html
For a graph $G$, we define a total $k$-labeling $\varphi$ as a combination of an edge labeling $\varphi_e:E(G)\rightarrow \{1,\,2,\,\ldots,\,k_e\}$ and a vertex labeling $\varphi_v:V(G)\rightarrow \{0,\,2,\,\ldots,\,2k_v\}$, where $k=\,\mbox{max}\, \{k_e,2k_v\}$. The total $k$-labeling $\varphi$ is called a vertex irregular reflexive $k$-labeling of $G$ if any pair of vertices $u$, $u'$ have distinct vertex weights $wt_{\varphi}(u)\neq wt_{\varphi}(u')$, where $wt_{\varphi}(u)=\varphi(u)+\sum_{uu'\in E(G)} \varphi(uu')$ for any vertex $u\in V(G)$. The smallest value of $k$ for which such a labeling exists is called the reflexive vertex strength of $G$, denoted by $rvs{(G)}$. In this paper, we present a new lower bound for the reflexive vertex strength of any graph. We investigate the exact values of the reflexive vertex strength of generalized friendship graphs, corona product of two paths, and corona product of a cycle with isolated vertices by referring to the lower bound. This study discovers some interesting open problems that are worth further exploration.Some algebraic properties of the subdivision graph of a graph
http://comb-opt.azaruniv.ac.ir/article_14540.html
Let $G=(V,E)$ be a connected graph with the vertex-set $V$ and &nbsp;the edge-set $E$. &nbsp; &nbsp;The subdivision graph $S(G)$ of the graph $G$ is obtained from $G$ by adding a vertex in the middle of every edge of $G$. &nbsp;In this paper, we investigate some properties of the graphs &nbsp;$S(G)$ and $L(S(G))$, where $L(S(G))$ is the line graph of $S(G)$. We will see that $S(G)$ and &nbsp;$L(S(G))$ &nbsp;inherit some &nbsp;properties of $G$. &nbsp; &nbsp;For instance, we show that if $G \ncong C_n$, then $Aut(G) \cong Aut(L(S(G)))$ (as abstract groups), where $C_n$ is the cycle of order $n$.1-Edge contraction: Total vertex stress and confluence number
http://comb-opt.azaruniv.ac.ir/article_14535.html
This paper introduces certain relations between $1$-edge contraction and the total vertex stress and the confluence number of a graph. A main result states that if a graph $G$ with $\zeta(G)=k\geq 2$ has an edge $v_iv_j$ and a $\zeta$-set $\mathcal{C}_G$ such that $v_i,v_j\in \mathcal{C}_G$ then, $\zeta(G/v_iv_j) = k-1$. In general, either $\mathcal{S}(G/e_i) \leq \mathcal{S}(G/e_j)$ or $\mathcal{S}(G/e_j) \leq \mathcal{S}(G/e_i)$ is true. This observation leads to an investigation into the question: for which edge(s) $e_i$ will $\mathcal{S}(G/e_i) = \max\{\mathcal{S}(G/e_j):e_j \in E(G)\}$ and for which edge(s) will $\mathcal{S}(G/e_j) = \min\{\mathcal{S}(G/e_\ell):e_\ell \in E(G)\}$?Triangular Tile Latching System
http://comb-opt.azaruniv.ac.ir/article_14470.html
A triangular tile latching system consists of a set $\Sigma$ of equilateral triangular tiles with at least one latchable side and an attachment rule which permits two tiles to get latched along a latchable side. In this paper we determine the language generated by a triangular tile latching system in terms of planar graphs.Some new families of generalized $k$-Leonardo and Gaussian Leonardo Numbers
http://comb-opt.azaruniv.ac.ir/article_14544.html
In this paper, we introduce a new family of the generalized $k$-Leonardo numbers and study their properties. We investigate the Gaussian Leonardo numbers and associated new families of these Gaussian forms. We obtain combinatorial identities like Binet formula, Cassini's identity, partial sum, etc. in the closed form. Moreover, we give various generating and exponential generating functions.Power Dominator Chromatic Numbers of Splitting Graphs of Certain Classes of Graphs
http://comb-opt.azaruniv.ac.ir/article_14513.html
Domination in graphs and coloring of graphs are two main areas of investigation in graph theory. Power domination is a variant of domination in graphs introduced in the study of the problem of monitoring an electric power system. Based on the notions of power domination and coloring of a graph, the concept of power dominator coloring of a graph was introduced. The minimum number of colors required for power dominator coloring of a graph $G$ is called the power dominator chromatic number $\chi_{pd}(G)$ of $G,$ which has been computed for some classes of graphs. Here we compute the power dominator chromatic number for splitting graphs of certain classes of graphs.Vector valued switching in signed graphs
http://comb-opt.azaruniv.ac.ir/article_14570.html
A signed graph is a graph with edges marked positive and negative; it is unbalanced if some cycle has negative sign product. We introduce the concept of vector valued switching function in signed graphs, which extends the concept of switching to higher dimensions. Using this concept, we define balancing dimension and strong balancing dimension for a signed graph, which can be used for a new classification of degree of imbalance of unbalanced signed graphs. We provide bounds for the balancing and strong balancing dimensions, and calculate these dimensions for some classes of signed graphs.A new construction for µ-way Steiner trades
http://comb-opt.azaruniv.ac.ir/article_14519.html
A $\mu$-way $(v,k,t)$ trade $T$ of volume $m$ consists of $\mu$ pairwise disjoint collections $T_1, \ldots ,T_{\mu}$, each of $m$ blocks of size $k$ such that for every $t$-subset of a $v$-set $V,$ the number of blocks containing this $t$-subset is the same in each $T_i$ for $1\leq i\leq \mu$. If any $t$-subset of the $v$-set $V$ occurs at most once in each $T_i$ for $1\leq i\leq \mu$, then $T$ is called a $\mu$-way $(v,k,t)$ Steiner trade. In 2016, it was proved that there exists a 3-way $(v,k,2)$ Steiner trade of volume $m$ when $12(k-1)\leq m$ for each $k$. Here we improve the lower bound to $8(k-1)$ for even $k$, by &nbsp;using a recursive construction.A study on structure of codes over $\mathbb Z_4+u\mathbb Z_4+v\mathbb Z_4 $
http://comb-opt.azaruniv.ac.ir/article_14548.html
We study $(1+2u+2v)$-constacyclic code over a semi-local ring $S=\mathbb Z_4+u\mathbb Z_4+v\mathbb Z_4$ with the condition $u^2=3u,v^2=3v$, and $uv=vu=0$, &nbsp;we show that &nbsp;$(1+2u+2v)$-constacyclic code over $S$ is equivalent to quasi-cyclic code over $\mathbb{Z}_4$ by using two new Gray maps from $S$ to $\mathbb{Z}_4.$ Also, for odd length $n$ we have defined a generating set for constacyclic codes over $S.$ Finally, we obtained some examples which are new to the data base [Database of $\mathbb{Z}_4$ codes [online]}, http://$\mathbb{Z}_4$ Codes.info(Accessed March 2, 2020)].The higher-order Sombor index
http://comb-opt.azaruniv.ac.ir/article_14623.html
Based on the geometric background of Sombor index and motivating by the higher order connectivity index and the Sombor index, we introduce the pathcoordinate of a path in a graph and a degree-point in a higher dimensional coordinate system, and define the higher order Sombor index of a graph. We first consider mathematical properties of the higher order Sombor index, show that the higher order connectivity index of a starlike tree is completely determined by its branches and that starlike trees with a given maximum degree which have the same higher order Sombor indices are isomorphic. Then, we determine the extremal values of the second order Sombor index for all trees with n vertices and characterize the corresponding extremal trees. Finally, the chemical importance of the second order Sombor index is investigated and it is shown that the new index is useful in predicting physicochemical properties with high accuracy compared to some well-established.On the Roman Domination Polynomials
http://comb-opt.azaruniv.ac.ir/article_14558.html
&lrm;&lrm;A Roman dominating function (RDF) on a graph $G$ is a function $ f:V(G)\to \{0,1,2\}$ &nbsp;satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. The weight of an RDF $f$ is the sum of the weights of the vertices under $f$. The Roman domination number, $\gamma_R(G)$ of $G$ is the minimum weight of an RDF in $G$. The Roman domination polynomial of a graph $G$ of order $n$ is the polynomial $RD(G,x)=\sum_{i=\gamma_R(G)}^{2n} d_R(G,i) x^{i}$, where $d_R(G,i)$ is the number of RDFs of $G$ with weight $i$. In this paper we prove properties of Roman domination polynomials and determine $RD(G,x)$ in several classes of graphs $G$ by new approaches. We also present bounds on the number of all Roman domination polynomials in a graph.Spectral determination of trees with large diameter and small spectral radius
http://comb-opt.azaruniv.ac.ir/article_14632.html
Yuan, Shao and Liu proved &nbsp;that the H-shape tree $H'_n =P_{1,2;n-3}^{1,n-6}$ minimizes the spectral radius among all graphs with order $n\geqslant 9$ and diameter $n-4$. In this paper, we achieve the spectral characterization of all graphs in the set $\mathscr{H}' = \{ H'_n\}_{n\geqslant 8}$. More precisely we show that $H'_n$ is determined by its spectrum if and only if $n \neq 8, 9,12$, and detect all cospectral mates of $H'_8$, $H'_9$ and $H'_{12}$. Divisibility between characteristic polynomials of graphs turns out to be an important tool to reach our goals.Monophonic Eccentric Domination in Graphs
http://comb-opt.azaruniv.ac.ir/article_14559.html
For any two vertices $u$ and $v$ in a connected graph $G,$ the monophonic distance $d_m(u,v)$ from $u$ to $v$ is defined as the length of a longest $u-v$ monophonic path in $G$. The monophonic eccentricity $e_m(v)$ of a vertex $v$ in $G$ is the maximum monophonic distance from $v$ to a vertex of $G$. &nbsp;A vertex $v$ in $G$ is a monophonic eccentric vertex of a vertex $u$ in $G$ if $e_m(u) = d_m(u,v)$. A set $S \subseteq V$ &nbsp;is a &nbsp;monophonic eccentric &nbsp;dominating $set$ if every vertex in $V-S$ has a monophonic eccentric vertex in $S$. The monophonic eccentric &nbsp;domination number $\gamma_{me}(G)$ is the &nbsp;cardinality of a minimum monophonic eccentric &nbsp;dominating set of $G$. We investigate some properties of monophonic eccentric &nbsp;dominating sets. Also, we determine the bounds of monophonic eccentric&nbsp; domination number and find the same for some standard graphs.Zero forcing number for Cartesian product of some graphs
http://comb-opt.azaruniv.ac.ir/article_14561.html
The zero forcing number of a graph $G$, denoted $Z(G)$, is a graph parameter &nbsp;which is based on a color change rule that describes how to color the vertices. Zero forcing is useful in several branches of science such as electrical engineering, computational complexity and quantum control. &nbsp;In this paper, we investigate the zero forcing number for Cartesian products of some graphs. The main contribution of this paper is to introduce a new presentation of the Cartesian product of two complete bipartite graphs and to obtain the zero forcing number of these graphs. &nbsp;We also introduce a purely graph theoretical method to prove $Z(K_n \Box K_m)=mn-m-n+2$.On $\gamma$-free, $\gamma$-totally-free and $\gamma$-fixed sets in graphs
http://comb-opt.azaruniv.ac.ir/article_14583.html
Let $G=(V,E)$ be a connected graph. A subset $S$ of $V$ is called a $\gamma$-free set if there exists a $\gamma$-set $D$ of $G$ such that $S \cap D= \emptyset$. If further the induced subgraph $H=G[V-S]$ is connected, then $S$ is called a&nbsp; $cc$-$\gamma$-free set of $G$. We use this concept to identify connected induced subgraphs $H$ of a given graph $G$ such that $\gamma(H) \leq \gamma(G)$. We also introduce the concept of $\gamma$-totally-free and $\gamma$-fixed sets and present several basic results on the corresponding parameters.&nbsp;Algebraic-based primal interior-point algorithms for stochastic infinity norm optimization
http://comb-opt.azaruniv.ac.ir/article_14581.html
We study the two-stage stochastic infinity norm optimization problem with recourse based on a commutative algebra. First, we explore and develop the algebraic structure of the infinity norm cone, and utilize it to compute the derivatives of the barrier recourse functions. Then, we prove that the barrier recourse functions and the composite barrier functions for this optimization problem are self-concordant families with reference to barrier parameters. These findings are used to develop interior-point algorithms based on primal decomposition for this class of stochastic programming problems. Our complexity results for the short- and long-step algorithms show that the dominant complexity terms are linear in the rank of the underlying cone. Despite the asymmetry of the infinity norm cone, we also show that the obtained complexity results match (in terms of rank) the best known results in the literature for other well-studied stochastic symmetric cone programs. Finally, we demonstrate the efficiency of the proposed algorithm by presenting some numerical experiments on both stochastic uniform facility location problems and randomly-generated problems.On graphs with integer Sombor indices
http://comb-opt.azaruniv.ac.ir/article_14584.html
&lrm;Sombor index of a graph $G$ is defined by $SO(G) = \sum_{uv \in E(G)} \sqrt{d^2_G(u)+d^2_G(v)}$, where $d_G(v)$ is the degree of the vertex $v$ of $G$. An $r$-degree graph is a graph whose degree sequence includes exactly $r$ distinctive numbers. In this article, we study $r$-degree connected graphs with integer Sombor index for $r \in \{5, 6, 7\}$. We show that: if $G$ is a 5-degree connected graph of order $n$ with integer Sombor index then $n \geq 50$ and the equality occurs if only if $G$ is a bipartite graph of size 420 with $SO(G) = 14830$; if $G$ is a 6-degree connected graph of order $n$ with integer Sombor index then $n \geq 75$ and the equality is established only for the bipartite graph of size $1080$; and if $G$ is a 7-degree connected graph of order $n$ with integer Sombor index then $n \geq 101$ and the equality is established only for the bipartite graph of size $1680$.Some Properties and Identities of Hyperbolic Generalized k-Horadam Quaternions and Octonions
http://comb-opt.azaruniv.ac.ir/article_14600.html
The aim of this paper is to introduce the hyperbolic generalized $k$-Horadam quaternions and octonions and investigate their algebraic properties. We present some properties and identities of these quaternions and octonions for generalized $k$-Horadam numbers. Moreover, we give some determinants related to the hyperbolic generalized $k$-Horadam quaternions and octonions. Finally, we evaluate its determinants through the Chebyshev polynomials of the second kind and give an illustrative example as well.Cost, Revenue and Profit Efficiency in multi-period network system: A DEA-R based Approach
http://comb-opt.azaruniv.ac.ir/article_14587.html
It has been proven that Data Envelopment Analysis is an efficient method to compare different decision making units with multiple inputs and outputs, but traditional Data Envelopment Analysis models suffers some difficulties: (a)- the inputs and outputs are not supposed to be given in terms of ratio. Thus, when the data are partially available, the decision maker will be unable to access missing data from the present data; (b) in measuring the efficiency of a set of decision making units for some periods, the conventional Data Envelopment Analysis based technique cannot handle the problem posed in a periodic form where the costs, profits and revenue efficiency of the main problems in the network structures are required. The contribution of this paper is four folded: (1) the cost, revenue and profit efficiency of each stages are calculated from the proposed method depends on the performance of the unit in both stages. (2) Our method evaluates the total cost, revenue and profit efficiency in a whole t(t=1,&hellip;,T) time periods derived from all periodic and every stage efficiency, (3) The proposed method in this study yields the efficiency measures deals with ratio data, (4) To elucidate the details of the proposed method, the proposed multi-period DEA-R method was employed to measure the efficiency of ten units in three separate time periods. Numerical examples are also provided to explain the presented methods.Game Chromatic Number of Honeycomb Related Networks
http://comb-opt.azaruniv.ac.ir/article_14591.html
Let $G$ be a simple connected graph having finite number of vertices (nodes). Let a coloring game is played on the nodes of $G$ by two players, Alice and Bob alternately assign colors to the nodes such that the adjacent nodes receive different colors with Alice taking first turn. Bob wins the game if he is succeeded to assign k distinct colors in the neighborhood of some vertex, where k is the available number of colors. Otherwise, Alice wins. The game chromatic number of G is the minimum number of colors that are needed for Alice to win this coloring game and is denoted by $\chi_{g}(G)$. In this paper, the game chromatic number $\chi_{g}(G)$ for some interconnecting networks such as infinite honeycomb network, elementary wall of infinite height and infinite octagonal network is determined. Also, the bounds for the game chromatic number $\chi_{g}(G)$ of infinite oxide network are explored.Some properties of star-perfect graphs
http://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.Strong domination number of some operations on a graph
http://comb-opt.azaruniv.ac.ir/article_14603.html
Let $G=(V(G),E(G))$ be a simple graph. A set $D\subseteq V(G)$ is a strong dominating set of $G$, if for every vertex $x\in V(G)\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $\deg(x)\leq \deg(y)$. The strong domination number $\gamma_{st}(G)$ is defined as the minimum cardinality of a strong dominating set. &nbsp;In this paper, we examine the effects on $\gamma_{st}(G)$ when $G$ is modified by operations on edge (or edges) of $G$.A note on the small quasi-kernels conjecture in digraphs
http://comb-opt.azaruniv.ac.ir/article_14608.html
A subset $K$ of vertices of digraph $D=(V(D),A(D))$ is a kernel if the following two conditions are fulfilled: (i) no two vertices of $K$ are connected by an arc in any direction and (ii) every vertex not in $K$ has an ingoing arc from some vertex in $K.$ A quasi-kernel of $D$ is a subset $Q$ of vertices satisfying condition (i) and furthermore every vertex can be reached in at most two steps from $Q.$ A vertex is source-free if it has at least one ingoing arc. In 1976, P.L. Erd&ouml;s and L.A. Sz&eacute;kely conjectured that every source-free digraph $D$ has a quasi-kernel of size at most $\left\vert V(D)\right\vert /2.$ Recently, this conjecture has been shown to be true by Allan van Hulst for digraphs having kernels. In this note, we provide a short and simple proof of van Hulst's result. We additionally characterize all source-free digraphs $D$ having kernels with smallest quasi-kernels of size $\left\vert V(D)\right\vert /2.$Bounds on Sombor index and inverse sum indeg (ISI) index of graph operations
http://comb-opt.azaruniv.ac.ir/article_14609.html
Let $ G $ be a graph with vertex set $ V(G) $ and edge set $ E(G) $. Denote by $ d_G(u) $ the degree of a vertex $ u \in V(G) $. The Sombor index of $ G $ is defined as $ SO(G) = \sum_{uv \in E(G)} \sqrt{d_u^2 + d_v^2} $, whereas, the inverse sum indeg $ (ISI) $ index is defined as $ ISI(G) = \sum_{uv \in E(G)} &nbsp; &nbsp;\frac{d_{u}d_{v}}{d_{u} + d_{v}}. $ In this paper, we compute the bounds in terms of maximum degree, minimum degree, order and size of the original graphs $ G $ and $ H $ for Sombor and $ ISI $ indices of several graph operations like corona product, cartesian product, strong product, composition and join of graphs.Characterization of Product Cordial Dragon Graphs
http://comb-opt.azaruniv.ac.ir/article_14610.html
The vertices of a graph are to be labelled with 0 or 1 such that each edge gets the label as the product of its end vertices. If the number of vertices labelled with 0's and 1's differ by at most one and if the number of edges labelled with 0's and 1's differ by at most by one, then the labelling is called product cordial labelling. Complete characterizations of product cordial dragon graphs are given. We also characterize dragon graphs whose line graphs are product cordial.On the complement of the intersection graph of subgroups of a group
http://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.On the Sombor Index of Sierpiński and Mycielskian Graphs
http://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 \(SO\left(\mu_m\left(K_n\right)\right)\).Some observations on Sombor coindex of graphs
http://comb-opt.azaruniv.ac.ir/article_14621.html
Let $G=(V,E)$, $V=\left\{ v_{1},v_{2},\ldots ,v_{n}\right\}$, be a simple graph of order $n$ and size $m$, without isolated vertices. The Sombor coindex of a graph $G$ is defined as &nbsp;$\overline{SO}(G)=\sum_{i\nsim j}\sqrt{d_i^2+d_j^2}$ , where $d_i= d(v_i)$ is a degree of vertex $v_i$, $i=1,2,\ldots , n$. In this paper we investigate a relationship &nbsp;between &nbsp;Sombor coindex and a number of other topological coindices.Well ve-covered graphs
http://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.Graphs with unique minimum edge-vertex dominating sets
http://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.Commuting graph of an aperiodic Brandt Semigroup
http://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 coherent configuration of circular-arc graphs
http://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.Finite Abelian Groups with Isomorphic Inclusion Graphs of Cyclic Subgroups
http://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.On Zero-Divisor Graph of the ring $\mathbb{F}_p+u\mathbb{F}_p+u^2 \mathbb{F}_p$
http://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 $\Gamma(R).$Antipodal Number of Cartesian Product of Complete Graphs with Cycles
http://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 $ac(K_r\square C_s)$.Vector valued switching in the products of signed graphs
http://comb-opt.azaruniv.ac.ir/article_14647.html
A signed graph is a graph whose edges are labeled either as positive or negative. The concepts of vector valued switching and balancing dimension of signed graphs were introduced by S. Hameed et al. In this paper, we deal with the balancing dimension of various products of signed graphs, namely the Cartesian product, the lexicographic product, the tensor product, and the strong product.Cliques in the extended zero-divisor graph of finite commutative rings
http://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.NP-completeness of some generalized hop and step domination parameters in graphs
http://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.The zero-divisor associate graph over a finite commutative ring
http://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.On the distance-transitivity of the folded hypercube
http://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.$k$-Secure Sets and $k$-Security Number of a Graph
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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.Sharp lower bounds on the metric dimension of circulant graphs
http://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 $t \ge 6$.On the ordering of the Randić index of unicyclic and bicyclic graphs
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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.)
http://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 $D_n$
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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.Signed total Italian $k$-domination in digraphs
http://comb-opt.azaruniv.ac.ir/article_14531.html
Let $k\ge 1$ be an integer, and let $D$ be a finite and simple digraph with vertex set $V(D)$. A signed total Italian $k$-dominating function (STIkDF) on a digraph $D$ is a function $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the conditions that (i) $\sum_{x\in N^-(v)}f(x)\ge k$ for each vertex $v\in V(D)$, where $N^-(v)$ consists of all vertices of $D$ from which arcs go into $v$, and (ii) each vertex $u$ with $f(u)=-1$ has an in-neighbor $v$ for which $f(v)=2$ or two in-neighbors $w$ and $z$ with $f(w)=f(z)=1$. The weight of an STIkDF $f$ is $\omega(f)=\sum_{v\in V(D)}f(v)$. The signed total Italian $k$-domination number $\gamma_{stI}^k(D)$ of $D$ is the minimum weight of an STIkDF on $D$. In this paper we initiate the study of the signed total Italian $k$-domination number of digraphs, and we &nbsp;present different bounds on $\gamma_{stI}^k(D)$. In addition, we determine the signed total Italian $k$-domination number of some classes of digraphs.A path-following algorithm for stochastic quadratically constrained convex quadratic programming in a Hilbert space
http://comb-opt.azaruniv.ac.ir/article_14543.html
We propose logarithmic-barrier decomposition-based interior-point algorithms for solving two-stage stochastic quadratically constrained convex quadratic programming problems in a Hilbert space. We prove the polynomial complexity of the proposed algorithms, and show that this complexity is independent on the choice of the Hilbert space, and hence it coincides with the best-known complexity estimates in the finite-dimensional case. We also apply our results on a concrete example from the stochastic control theory.Maximal outerplanar graphs with semipaired domination number double the domination number
http://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
http://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
http://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.