Communications in Combinatorics and Optimization
http://comb-opt.azaruniv.ac.ir/
Communications in Combinatorics and Optimizationendaily1Sun, 31 Dec 2023 00:00:00 +0330Sun, 31 Dec 2023 00:00:00 +0330Time-subinterval shifting in zero-sum games played in staircase-function finite and uncountably infinite spaces
http://comb-opt.azaruniv.ac.ir/article_14517.html
A tractable and efficient method of solving zero-sum games played in staircase-function finite spaces is presented, where the possibility of varying the time interval on which the game is defined is considered. The time interval can be narrowed by an integer number of time subintervals and still the solution is obtained by stacking solutions of smaller-sized matrix games, each defined on a subinterval where the pure strategy value is constant. The stack is always possible, even when only time is discrete and the set of pure strategy possible values is uncountably infinite. So, the solution of the initial discrete-time staircase-function zero-sum game can be obtained by stacking the solutions of the ordinary zero-sum games defined on rectangle, whichever the time interval is. Any combination of the solutions of the subinterval games is a solution of the initial zero-sum game.Domination parameters of the splitting graph of a graph
http://comb-opt.azaruniv.ac.ir/article_14420.html
Let $G=(V,E)$ be a graph of order $n$ and size $m.$ The graph $Sp(G)$ obtained from $G$ by adding a new vertex $v'$ for every vertex $v\in V$ and joining $v'$ to all neighbors of $v$ in $G$ is called the splitting graph of $G.$ In this paper, we determine the domination number, the total domination number, connected domination number, paired domination number and independent domination number for the splitting graph $Sp(G).$The Cartesian product of wheel graph and path graph is antimagic
http://comb-opt.azaruniv.ac.ir/article_14424.html
Suppose each edge of a simple connected undirected graph is given a unique number from the numbers $1, 2, \dots, $q$, where $q$ is the number of edges of that graph. Then each vertex is labelled with sum of the labels of the edges incident to it. If no two vertices have the same label, then the graph is called an antimagic graph. We prove that the Cartesian product of wheel graph and path graph is antimagic.Independent Italian bondage of graphs
http://comb-opt.azaruniv.ac.ir/article_14568.html
An independent Italian dominating function (IID-function) on a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2\}$ satisfying the conditions that (i) $\sum_{u\in N(v)}f(u)\geq2$ when $f(v)=0$, and (ii) the set of all vertices assigned non-zero values under $f$ is independent. The weight of an IID-function is the sum of its function values over all vertices, and the independent Italian domination number $i_{I}(G)$ of $G$ is the minimum weight of an IID-function on $G$. In this paper, we initiate the study of the independent Italian bondage number $b_{iI}(G)$ of a graph $G$ having at least one component of order at least three, defined as the smallest size of a set of edges of $G$ whose removal from $G$ increases $i_{I}(G)$. We show that the decision problem associated with the independent Italian bondage problem is NP-hard for arbitrary graphs. Moreover, various upper bounds on $b_{iI}(G)$ are established as well as exact values on it for some special graphs. In particular, for trees $T$ of order at least three, it is shown that $b_{iI}(T)\leq2$.On signs of several Toeplitz--Hessenberg determinants whose elements contain central Delannoy numbers
http://comb-opt.azaruniv.ac.ir/article_14432.html
In the paper, by virtue of Wronski's formula and Kaluza's theorem for the power series and its reciprocal, and with the aid of the logarithmic convexity of a sequence constituted by central Delannoy numbers, the authors present negativity of several Toeplitz--Hessenberg determinants whose elements contain central Delannoy numbers and combinatorial numbers.Linear-time construction of floor plans for plane triangulations
http://comb-opt.azaruniv.ac.ir/article_14427.html
This paper focuses on a novel approach for producing a floor plan (FP), either a rectangular (RFP) or an orthogonal (OFP) based on the concept of orthogonal drawings, which satisfies the adjacency relations given by any bi-connected plane triangulation $G$.&nbsp; &nbsp; &nbsp;Previous algorithms for constructing a FP are primarily restricted to the cases given below:&nbsp; &nbsp; &nbsp;\begin{enumerate}[(i)]&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\item A bi-connected plane triangulation without separating triangles (STs) and with at most 4 corner implying paths (CIPs), known as properly triangulated planar graph (PTPG).&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\item A bi-connected plane triangulation with an exterior face of length 3 and no CIPs, known as maximal planar graph (MPG).&nbsp; &nbsp; &nbsp;\end{enumerate}&nbsp; &nbsp; &nbsp;The FP obtained in the above two cases is a RFP or an OFP respectively. In this paper, we present the construction of a FP (RFP if exists, else an OFP), for a bi-connected plane triangulation $G$ in linear-time.On local antimagic chromatic number of various join graphs
http://comb-opt.azaruniv.ac.ir/article_14428.html
A local antimagic edge labeling of a graph $G=(V,E)$ is a bijection $f:E\rightarrow\{1,2,\dots,|E|\}$ such that the induced vertex labeling $f^+:V\rightarrow \mathbb{Z}$ given by $f^+(u)=\sum f(e),$ where the summation runs over all edges $e$ incident to $u,$ has the property that any two adjacent vertices have distinct labels. A graph $G$ is said to be locally antimagic if it admits a local antimagic edge labeling. The local antimagic chromatic number $\chi_{la}(G)$ is the minimum number of distinct induced vertex labels over all local antimagic &nbsp;labelings of $G.$ In this paper we obtain sufficient conditions under which $\chi_{la}(G\vee H),$ where $H$ is either a cycle or the empty graph $O_n=\overline{K_n},$ satisfies a sharp upper bound. Using this we determine the value of $\chi_{la}(G\vee H)$ for many wheel related graphs $G.$Some properties of the essential annihilating-ideal graph of commutative rings
http://comb-opt.azaruniv.ac.ir/article_14448.html
Let $\mathcal{S}$ be a commutative ring with unity and $A(\mathcal{S})$ denotes the set of annihilating-ideals of $\mathcal{S}$. The essential annihilating-ideal graph of $\mathcal{S}$, denoted by $\mathcal{EG}(\mathcal{S})$, is an undirected graph with $A^*(\mathcal{S})$ as the set of vertices and &nbsp; for distinct $\mathcal{I}, \mathcal{J} \in A^*(\mathcal{S})$, $\mathcal{I} \sim \mathcal{J}$ is an edge if and only if $Ann(\mathcal{IJ}) \leq_e \mathcal{S}$. In this paper, we classify the Artinian rings $\mathcal{S}$ for which $\mathcal{EG}(\mathcal{S})$ is projective. We also discuss the coloring of $\mathcal{EG}(\mathcal{S})$. Moreover, we discuss the domination number of $\mathcal{EG}(\mathcal{S})$.Cycle transit function and betweenness
http://comb-opt.azaruniv.ac.ir/article_14433.html
Transit functions are introduced to study betweenness, intervals and convexity in an axiomatic setup on graphs and other discrete structures. Prime example of a transit function on graphs is the well studied interval function of a connected graph. In this paper, we study the Cycle transit function $\mathcal{C}( u,v)$ on graphs which is a transit function derived from the interval function. We study the betweenness properties and also characterize graphs in which the cycle transit function coincides with the interval function. We also characterize graphs where $|\mathcal{C}( u,v)\cap \mathcal{C}( v,w) \cap \mathcal{C}( u,w)|\le 1$ as an analogue of median graphs.On several new closed-form evaluations for the generalized hypergeometric functions
http://comb-opt.azaruniv.ac.ir/article_14434.html
The main objective of this paper is to establish as many as thirty new closed-form evaluations of the generalized hypergeometric function $_{q+1}F_q(z)$ for $q= 2, 3$. This is achieved by means of separating the generalized hypergeometric function $_{q+1}F_q(z)$ for $q=1, 2, 3$ into even and odd components together with the use of several known infinite series involving reciprocal of the non-central binomial coefficients obtained earlier by L. Zhang and W. Ji.Lower bound on the KG-Sombor index
http://comb-opt.azaruniv.ac.ir/article_14580.html
&lrm;In 2021, a novel degree-based topological index was introduced by Gutman, called the &nbsp;Sombor index. Recently Kulli and Gutman introduced a vertex-edge variant of the Sombor index, is caled KG-Sombor index. In this paper, we establish lower bound on the KG-Sombor index and determine the extremal trees achieve this bound.Roman domination number of signed graphs
http://comb-opt.azaruniv.ac.ir/article_14443.html
A function $f:V\rightarrow \{0,1,2\}$ on a signed graph $S=(G,\sigma)$ &nbsp;where $G = (V,E)$ is a Roman dominating function(RDF) if $f(N[v]) = f(v) + \sum_{u \in N(v)} \sigma(uv)f(u) \geq 1$ for all $v\in V$ and for each vertex $v$ with $f(v)=0$ there is a vertex $u$ in $N^+(v)$ such that $f(u) = 2$. The weight of an RDF $f$ is given by $\omega(f) =\sum_{v\in V}f(v)$ and the minimum weight among all the RDFs on $S$ is called the Roman domination number $\gamma_R(S)$. Any RDF on $S$ with the minimum weight is known as a $\gamma_R(S)$-function. In this article we obtain certain bounds for $ \gamma_{R} $ and characterise the signed graphs attaining small values for $ \gamma_R. $Uniqueness of rectangularly dualizable graphs
http://comb-opt.azaruniv.ac.ir/article_14444.html
A generic rectangular partition &nbsp; is a partition of a rectangle into a finite number of rectangles provided that no four of them meet at a point. &nbsp;A graph $\mathcal{H}$ is called &nbsp;dual of a plane graph $\mathcal{G}$ if there is one$-$to$-$one correspondence between the vertices of $\mathcal{G}$ and the regions of $\mathcal{H}$, and &nbsp;two vertices of $\mathcal{G}$ are adjacent if and only if the corresponding regions of $\mathcal{H}$ are adjacent. A plane graph is a&nbsp; rectangularly dualizable graph &nbsp;if its dual &nbsp;can be embedded as a &nbsp;rectangular partition. &nbsp; A rectangular dual&nbsp; $\mathcal{R}$ of a plane graph $\mathcal{G}$ is a partition of a &nbsp;rectangle &nbsp;into $n-$rectangles such that (i) no four rectangles of $\mathcal{R}$ meet at a point, (ii) &nbsp;rectangles in &nbsp;$\mathcal{R}$ are mapped to vertices of $\mathcal{G}$, &nbsp;and (iii) &nbsp;two rectangles in $\mathcal{R}$ share a common boundary segment if and only if the corresponding vertices are adjacent in $\mathcal{G}$. In this paper, we derive a necessary and sufficient &nbsp;for a rectangularly dualizable graph $\mathcal{G}$ to admit a unique rectangular dual upto combinatorial &nbsp;equivalence. Further we show that $\mathcal{G}$ always admits &nbsp; a slicible as well as an area$-$universal &nbsp;rectangular dual.Bounds of point-set domination number
http://comb-opt.azaruniv.ac.ir/article_14445.html
A subset $D$ of the vertex set $V(G)$ in a graph $G$ is a point-set dominating set (or, in short, psd-set) of $G$ if for every set $S\subseteq V- D$, there exists a vertex $v\in D$ such that the induced subgraph $\langle S\cup \{v\}\rangle$ is connected. &nbsp;The minimum cardinality of a psd-set of $G$ is called the point-set domination number of $G$. In this paper, we establish two sharp lower bounds for point-set domination number of a graph in terms of its diameter and girth. We characterize graphs for which lower bound of point set domination number is attained in terms of its diameter. We also establish an upper bound and give some classes of graphs which attains the upper bound of point set domination number.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.Some lower bounds on the Kirchhoff index
http://comb-opt.azaruniv.ac.ir/article_14457.html
Let $G=(V,E)$, $V=\{v_1,v_2,\ldots,v_n\}$, $E=\{e_1,e_2,\ldots, e_m\}$, be a simple graph of order $n\ge 2$ and size $m$ without isolated vertices. Denote with $\mu_1\ge \mu_2\ge \cdots \ge \mu_{n-1}&gt;\mu_n=0$ the Laplacian eigenvalues of $G$. The Kirchhoff index of a graph $G$, &nbsp;defined in terms of Laplacian eigenvalues, is given as $Kf(G) = n \sum_{i=1}^{n-1}\frac{1}{\mu_i}$. Some new lower bounds on $Kf(G)$ are obtained.Bounds of Sombor Index for Corona Products on R-Graphs
http://comb-opt.azaruniv.ac.ir/article_14459.html
Operations in the theory of graphs has a substantial influence in the analytical and factual dimensions of the domain. In the realm of chemical graph theory, topological descriptor serves as a comprehensive graph invariant linked with a specific molecular structure. The study on the Sombor index is initiated recently by Ivan Gutman. The triangle parallel graph comprises of the edges of subdivision graph along with the edges of the original graph. In this paper, we make use of combinatorial inequalities related with the vertices, edges and the neighborhood concepts as well as the other topological descriptors in the computations for the determination of bounds of Sombor index for certain corona products involving the triangle parallel graph.Chromatic Transversal Roman Domination in Graphs
http://comb-opt.azaruniv.ac.ir/article_14461.html
For a graph $G$ with chromatic number $k$, a dominating set $S$ of $G$ is called a chromatic-transversal dominating set (ctd-set) if $S$ intersects every color class of any $k$-coloring of $G$. &nbsp;The minimum cardinality of a ctd-set of $G$ is called the {\em chromatic transversal domination number} of $G$ and is denoted by $\gamma_{ct}(G)$. &nbsp;A {\em Roman dominating function} (RDF) in a graph $G$ is a function $f : V(G) \to \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. &nbsp;The weight of a Roman dominating function is the value $w(f) = \sum_{u \in V} f(u)$. &nbsp;The minimum weight of a Roman dominating function of a graph $G$ is called the {\em Roman domination number} of $G$ and is denoted by $\gamma_R(G)$. &nbsp;The concept of {\em chromatic transversal domination} is extended to Roman domination as follows: &nbsp; For a graph $G$ with chromatic number $k$, a {\em Roman dominating function} $f$ is called a {\em chromatic-transversal Roman dominating function} (CTRDF) if the set of all vertices $v$ with $f(v) &gt; 0$ intersects every color class of any $k$-coloring of $G$. &nbsp;The minimum weight of a chromatic-transversal Roman dominating function of a graph $G$ is called the {\em chromatic-transversal Roman domination number} of $G$ and is denoted by $\gamma_{ctR}(G)$. &nbsp;In this paper a study of this parameter is initiated.On chromatic number and clique number in k-step Hamiltonian graphs
http://comb-opt.azaruniv.ac.ir/article_14462.html
A graph $G$ of order $n$ is called $k-$step Hamiltonian for $k\geq 1$ if we can label the vertices of $G$ as $v_1,v_2,\ldots,v_n$ such that $d(v_n,v_1)=d(v_i,v_{i+1})=k$ for $i=1,2,\ldots,n-1$. The (vertex) chromatic number of a graph $G$ is the minimum number of colors needed to color the vertices of $G$ so that no pair of adjacent vertices receive the same color. The clique number of $G$ is the maximum cardinality of a set of pairwise adjacent vertices in $G$. In this paper, we study the chromatic number and the clique number in $k-$step Hamiltonian graphs for $k\geq 2$. We present upper bounds for the chromatic number in $k-$step Hamiltonian graphs and give characterizations of graphs achieving the equality of the bounds. We also present an upper bound for the clique number in $k-$step Hamiltonian graphs and characterize graphs achieving equality of the bound.On Equitable Near Proper Coloring of Graphs
http://comb-opt.azaruniv.ac.ir/article_14463.html
A defective vertex coloring of a graph is a coloring in which some adjacent vertices may have the same color. An edge whose adjacent vertices have the same color is called a bad edge. A defective coloring of a graph $G$ with minimum possible number of bad edges in $G$ is known as a near proper coloring of $G$. &nbsp;In this paper, we introduce the notion of equitable near proper coloring of graphs and determine the minimum number of bad edges obtained from an equitable near proper coloring of some graph classes.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.Multiplicative Zagreb indices of trees with given domination number
http://comb-opt.azaruniv.ac.ir/article_14468.html
In [On extremal multiplicative Zagreb indices of trees with given domination number, Applied Mathematics and Computation 332 (2018), 338--350] Wang et al. presented bounds on the multiplicative Zagreb indices of trees with given domination number. We fill in the gaps in their proofs of Theorems 3.1 and 3.3 and we correct Theorem 3.3.Tetravalent half-arc-transitive graphs of order $12p$
http://comb-opt.azaruniv.ac.ir/article_14469.html
A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not its arc set. In this paper, we study &nbsp;all tetravalent half-arc-transitive graphs of order $12p$, &nbsp;where $p$ is a prime.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.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.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.Further results on the j-independence number of graphs
http://comb-opt.azaruniv.ac.ir/article_14479.html
In a graph $G$ of minimum degree $\delta$ and maximum degree $\Delta$, a subset $S$ of vertices of $G$ is $j$-independent, for some positive integer $j,$ if every vertex in $S$ has at most $j-1$ neighbors in $S$. The $j$-independence number $\beta_{j}(G)$ is the maximum cardinality of a $j$-independent set of $G$. We first establish an inequality between $\beta_{j}(G)$ and $\beta_{\Delta}(G)$ for $1\leq j\leq\delta-1$. Then we characterize all graphs $G$ with $\beta_{j}(G)=\beta_{\Delta}(G)$ for $j\in\{1,\dots,\Delta-1\}$, where the particular cases $j=1,2,\delta-1$ and$\delta$ are well distinguished.Maximizing the indices of a class of signed complete graphs
http://comb-opt.azaruniv.ac.ir/article_14484.html
The index of a signed graph is the largest eigenvalue of its adjacency matrix. Let $\mathfrak{U}_{n,k,4}$ be the set of all signed complete graphs of order $n$ whose negative edges induce a unicyclic graph of order $k$ and girth at least $4$. In this paper, we identify the signed graphs achieving the maximum index in the class $\mathfrak{U}_{n,k,4}$.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.Extremal Kragujevac trees with respect to Sombor indices
http://comb-opt.azaruniv.ac.ir/article_14503.html
The concept of the Sombor indices of a graph was introduced by Gutman. A vertex-edge variant of the Sombor index of graphs is called the KG-Sombor index. &nbsp;Recently, the Sombor and &nbsp;KG-Sombor indices of Kragujevac trees were studied, and the extremal Kragujevac trees with respect to these indices were empirically &nbsp;determined. &nbsp; Here we give analytical proof of the results.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 $(|V|^{O(\log\log|V |)})$.A Counter example 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 $G$.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.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.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.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.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.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.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.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 $\mathcal{S}(G/e_j) = \min\{\mathcal{S}(G/e_\ell):e_\ell \in E(G)\}$?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 $n$.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 $10$.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.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.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.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)].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.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.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 $Z(K_n \Box K_m)=mn-m-n+2$.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.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.The energy and edge energy of some Cayley graphs on the abelian group $\mathbb{Z}_{n}^{4}$
http://comb-opt.azaruniv.ac.ir/article_14582.html
Let $G=(V, E)$ be a simple graph such that $\lambda_1, \ldots, \lambda_n$ be the eigenvalues of $G$. The energy of graph $G$ is denoted by $E(G)$ and is defined as $E(G)=\sum_{i=1}^{n}|\lambda_{i}|$. The edge energy of $G$ is the energy of line graph $G$. In this paper, we investigate the energy and edge energy for two Cayley graphs on the abelian group $\mathbb{Z}_{n}^{4}$, namely, the Sudoku graph and the positional Sudoku graph. Also, we obtain graph energy and edge energy of the complement of these two graphs.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;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 $1680$.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 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.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 \(SO\left(\mu_m\left(K_n\right)\right)\).Simultaneous coloring of vertices and incidences of hypercubes
http://comb-opt.azaruniv.ac.ir/article_14618.html
An element $i=(v,e)$ of a graph $G$ is called &nbsp;an incidence of $G$, if $v\in V(G)$, $e\in E(G)$ and $v\in e$. The simultaneous coloring of vertices and incidences of a graph is coloring &nbsp;the vertices and incidences of the graph properly at the same time such that any two adjacent or incident elements receive distinct colors. In this paper, we investigate the simultaneous coloring of vertices and incidences of hypercubes.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.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.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.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 that the H-shape tree H&#039;n = P^{1,n&minus;6}_{1,2;n&minus;3} minimizes the spectral radius among all graphs with order n ⩾ 9 and diameter n&minus;4. In this paper, we achieve the spectral characterization of all graphs in the set H &prime; = {H&prime;n }n⩾8. More precisely we show that H&prime;n is determined by its spectrum if and only if n \not= 8, 9, 12, and detect all cospectral mates of H&#039;8 , H&#039;9 and H&#039;12. Divisibility between characteristic polynomials of graphs turns out to be an important tool to reach our goals.