Communications in Combinatorics and Optimization
http://comb-opt.azaruniv.ac.ir/
Communications in Combinatorics and Optimizationendaily1Thu, 01 Jun 2023 00:00:00 +0430Thu, 01 Jun 2023 00:00:00 +0430Pareto-efficient strategies in 3-person games played with staircase-function strategies
http://comb-opt.azaruniv.ac.ir/article_14354.html
A tractable method of solving 3-person games in which players&rsquo; pure strategies are staircase functions is suggested. The solution is meant to be Pareto-efficient. The method considers any 3-person staircase-function game as a succession of 3-person games in which strategies are constants. For a finite staircase-function game, each constant-strategy game is a trimatrix game whose size is likely to be relatively small to solve it in a reasonable time. It is proved that any staircase-function game has a single Pareto-efficient situation if every constant-strategy game has a single Pareto-efficient situation, and vice versa. Besides, it is proved that, whichever the staircase-function game continuity is, any Pareto-efficient situation of staircase function-strategies is a stack of successive Pareto-efficient situations in the constant-strategy games. If a staircase-function game has two or more Pareto-efficient situations, the best efficient situation is one which is the farthest from the triple of the most unprofitable payoffs. In terms of 0-1-standardization, the best efficient situation is the farthest from the triple of zero payoffs.New bounds on Sombor index
http://comb-opt.azaruniv.ac.ir/article_14356.html
The Sombor index of the graph $G$ is a degree based topological index, defined as $SO = \sum_{uv \in \mathbf E(G)}\sqrt{d_u^2+d_v^2}$, where $d_u$ is the degree of the vertex $u$, and $\mathbf E(G)$ is the edge set of $G$. Bounds on $SO$ are established in terms of graph energy, size of minimum vertex cover, matching number, and induced matching number.Line signed graph of a signed unit graph of commutative rings
http://comb-opt.azaruniv.ac.ir/article_14357.html
In this paper we characterize the commutative rings with unity for which line signed graph of signed unit graph is balanced and consistent. To do this, first we derive some sufficient conditions for balance and consistency of signed unit graphs. The results have been demonstrated with ample number of examples.Unit $\mathbb{Z}_q$-Simplex codes of type α and zero divisor $\mathbb{Z}_q$-Simplex codes
http://comb-opt.azaruniv.ac.ir/article_14358.html
In this paper, we have punctured unit $\mathbb{Z}_q$-Simplex code &nbsp;and constructed a new code called unit $\mathbb{Z}_q$-Simplex code of type $\alpha$. In particular, we find the parameters of &nbsp;these codes and have proved that it is an $\left[\phi(q)+2, ~\hspace{2pt} 2, ~\hspace{2pt} \phi(q)+2 - \frac{\phi(q)}{\phi(p)}\right]$ $\mathbb{Z}_q$-linear code $\text{if} ~ k=2$ and $\left[\frac{\phi(q)^k-1}{\phi(q)-1}+\phi(q)^{k-2}, ~k,~ \frac{\phi(q)^k-1} {\phi(q)-1}+\phi(q)^{k-2}-\left(\frac{\phi(q)}{\phi(p)}\right)\left(\frac{\phi(q)^{k-1}-1}{\phi(q)-1}+\phi(q)^{k- 3}\right)\right]$ $\mathbb{Z}_q$-linear code if $k \geq 3, $ where $p$ is the smallest prime divisor of $q.$&nbsp; For $q$ is a prime power and rank $k=3,$ we have given the &nbsp;weight distribution of &nbsp;unit $\mathbb{Z}_q$-Simplex codes&nbsp; of type $\alpha$. Also, we have introduced some new code from &nbsp;$\mathbb{Z}_q$-Simplex code called zero divisor $\mathbb{Z}_q$-Simplex code and proved that it is an $\left[ \frac{\rho^k-1}{\rho-1}, \hspace{2pt} k, \hspace{2pt} \frac{\rho^k-1}{\rho-1}-\left(\frac{\rho^{(k-1)}-1}{\rho-1}\right)\left(\frac{q}{p}\right) \right]$ $\mathbb{Z}_{q}$-linear code, where $\rho = q-\phi(q)$ and $p$ is the smallest prime divisor of $q.$ Further, we obtain &nbsp;weight distribution of&nbsp; zero divisor $\mathbb{Z}_q$-Simplex code for rank $k=3$ and $q$ is a prime power.Roman domination in signed graphs
http://comb-opt.azaruniv.ac.ir/article_14371.html
Let $S = (G,\sigma)$ be a signed graph. A function $f: V \rightarrow \{0,1,2\}$ is a Roman dominating function on $S$ if $(i)$ for each $v \in V,$ $f(N[v]) = f(v) + \sum_{u \in N(v)} \sigma(uv ) f(u) \geq 1$ and $(ii)$ for each vertex $ v $ with $ f(v) = 0 $ there exists a vertex $u \in N^+(v)$ such that $f(u) = 2.$ In this paper we initiate a study on Roman dominating function on signed graphs. We characterise the signed paths, cycles and stars that admit a Roman dominating function.Cop-edge critical generalized Petersen and Paley graphs
http://comb-opt.azaruniv.ac.ir/article_14372.html
Cop Robber game is a two player game played on an undirected graph. In this game, the cops try to capture a robber moving on the vertices of the graph. The cop number of a graph is the least number of cops needed to guarantee that the robber will be caught. We study textit{cop-edge critical} graphs, i.e. graphs $G$ such that for any edge $e$ in $E(G)$ either $c(G-e)&lt; c(G)$ or $c(G-e)&gt;c(G)$. In this article, we study the edge criticality of generalized Petersen graphs and Paley graphs.&nbsp;More on the bounds for the skew Laplacian energy of weighted digraphs
http://comb-opt.azaruniv.ac.ir/article_14373.html
Let $\mathscr{D}$ be a simple connected digraph with $n$ vertices and $m$ arcs and let $W(\mathscr{D})=\mathscr{D},w)$ be the weighted digraph corresponding to $\mathscr{D}$, where the weights are taken from the set of non-zero real numbers. Let $nu_1,nu_2, \dots,nu_n$ be the eigenvalues of the skew Laplacian weighted matrix $\widetilde{SL}W(\mathscr{D})$ of the weighted digraph $W(\mathscr{D})$. In this paper, we discuss the skew Laplacian energy $\widetilde{SLE}W(\mathscr{D})$ of weighted digraphs and obtain the skew Laplacian energy of the weighted star $W(\mathscr{K}_{1, n})$ for some fixed orientation to the weighted arcs. We obtain lower and upper bounds for $\widetilde{SLE}W(\mathscr{D})$ and show the existence of weighted digraphs attaining these bounds.&nbsp;A lower bound for the second Zagreb index of trees with given Roman domination number
http://comb-opt.azaruniv.ac.ir/article_14376.html
For a (molecular) graph, the second Zagreb index $M_2(G)$ is equal to the sum of the products of the degrees of pairs of adjacent vertices. Roman dominating function $RDF$ of $G$ is a function $f:V(G)\rightarrow \{0,1,2\}$ satisfying the condition that every vertex with label 0 is adjacent to a vertex with label 2. The weight of an $RDF$ $f$ is $w(f)=\sum_{v\in V(G)} f(v)$. The Roman domination number of $G$, denoted by $\gamma_R (G)$, is the minimum weight among all RDF in $G$. In this paper, we present a lower bound on the second Zagreb index of trees with $n$ vertices and Roman domination number and thus settle one problem given in [On the Zagreb indices of graphs with given Roman domination number, Commun. Comb. Optim. DOI: 10.22049/CCO.2021.27439.1263 (article in press)].A study on graph topology
http://comb-opt.azaruniv.ac.ir/article_14384.html
The concept of topology defined on a set can be extended to the field of graph theory by defining the notion of graph topologies on graphs where we consider a collection of subgraphs of a graph $G$ in such a way that this collection satisfies the three conditions stated similarly to that of the three axioms of point-set topology. This paper discusses an introduction and basic concepts to the graph topology. A subgraph of $G$ is said to be open if it is in the graph topology $\mathscr{T}_G$. The paper also introduces the concept of the closed graph and the closure of graph topology in graph topological space using the ideas of decomposition-complement and neighborhood-complement.2S3 transformation for Dyadic fractions in the interval (0, 1)
http://comb-opt.azaruniv.ac.ir/article_14386.html
The $2S3$ transformation, which was first described for positive integers, has been defined for dyadic rational numbers in the open interval $(0,1)$ &nbsp;in this study. &nbsp;The set of dyadic rational numbers &nbsp;is a Pr&uuml;fer 2-group. For the dyadic $2S3$ transformation $T_{ds}(x)$, the restricted multiplicative and additive properties have been established. Graph parameters are used to generate more combinatorial outcomes for these properties. The relationship between the SM dyadic sum graph's automorphism group and the symmetric group has been investigated.Coalition Graphs
http://comb-opt.azaruniv.ac.ir/article_14431.html
A coalition in a graph $G = (V, E)$ consists of two disjoint sets $V_1$ and $V_2$ of vertices, such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1 \cup V_2$ is a dominating set of $G$. A coalition partition in a graph $G$ of order $n = |V|$ is a vertex partition $\pi = {V_1, V_2, \ldots, V_k}$ such that every set $V_i$ either is a dominating set consisting of a single vertex of degree $n-1$, or is not a dominating set but forms a coalition with another set $V_j$. Associated with every coalition partition $\pi$ of a graph $G$ is a graph called the coalition graph of $G$ with respect to $\pi$, denoted $CG(G,\pi)$, the vertices of which correspond one-to-one with the sets $V_1, V_2, \ldots, V_k$ of $\pi$ and two vertices are adjacent in $CG(G,\pi)$ if and only if their corresponding sets in $\pi$ form a coalition. In this paper, we initiate the study of coalition graphs and we show that every graph is a coalition graph.Normalized distance Laplacian matrices for signed graphs
http://comb-opt.azaruniv.ac.ir/article_14391.html
In this paper, we introduce the notion of normalized distance Laplacian matrices for signed graphs corresponding to the two signed distances defined for signed graphs. We characterize balance in signed graphs using these matrices and compare the normalized distance Laplacian spectral radius of signed graphs with that of all-negative signed graphs. Also we characterize the signed graphs having maximum normalized distance Laplacian spectral radius.Outer-independent total 2-rainbow dominating functions in graphs
http://comb-opt.azaruniv.ac.ir/article_14401.html
Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. An {outer-independent total $2$-rainbow dominating function of a graph $G$ is a function $f$ from $V(G)$ to the set of all subsets of $\{1,2\}$ such that the following conditions hold: (i) for any vertex $v$ with $f(v)=\emptyset$ we have $\bigcup_{u\in N_G(v)} f(u)=\{1,2\}$, (ii) the set of all vertices $v\in V(G)$ with $f(v)=\emptyset$ is independent and (iii) $\{v\mid f(v)\neq\emptyset\}$ has no isolated vertex. The outer-independent total $2$-rainbow domination number of $G$, denoted by ${\gamma}_{oitr2}(G)$, is the minimum value of $\omega(f)=\sum_{v\in V(G)} |f(v)|$ over all such functions $f$. In this paper, we study the outer-independent total $2$-rainbow domination number of $G$ and classify all graphs with outer-independent total $2$-ainbow domination number belonging to the set $\{2,3,n\}$. Among other results, we present some sharp bounds concerning the invariant.Signed total Italian domination in digraphs
http://comb-opt.azaruniv.ac.ir/article_14408.html
Let $D$ be a finite and simple digraph with vertex set $V(D)$. A signed total Italian dominating function (STIDF) on a digraph $D$ is a function $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the conditions that (i) $\sum_{x\in N^-(v)}f(x)\ge 1$ for each $v\in V(D)$, where $N^-(v)$ consists of all vertices of $D$ from which arcs go into $v$, and (ii) every vertex $u$ for which $f(u)=-1$ 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 &nbsp;STIDF $f$ is $\sum_{v\in V(D)}f(v)$. The signed total Italian domination number $\gamma_{stI}(D)$ of $D$ is the minimum weight of an STIDF on $D$. In this paper we initiate the study of the signed total Italian domination number of digraphs, and we &nbsp;present different bounds on $\gamma_{stI}(D)$. In addition, we determine the signed total Italian domination number of some classes of digraphs.Weak Roman domination stable graphs upon edge-addition
http://comb-opt.azaruniv.ac.ir/article_14410.html
A Roman dominating function (RDF) on a graph $G$ is a function $f: V(G) \to \{0, 1, 2\}$ such that every vertex with label 0 has a neighbor with label 2. A vertex $u$ with $f(u)=0$ is said to be undefended if it is not adjacent to a vertex with $f(v)&gt;0$. The function $f:V(G) \to \{0, 1, 2\}$ is a weak Roman dominating function (WRDF) if each vertex $u$ with $f(u) = 0$ is adjacent to a vertex $v$ with $f(v) &gt; 0$ such that the function $f^{\prime}: V(G) \to \{0, 1, 2\}$ defined by $f^{\prime}(u) = 1$, $f^{\prime}(v) = f(v) - 1$ and $f^{\prime}(w) = f(w)$ if $w \in V - \{u, v\}$, has no undefended vertex. A graph $G$ is said to be Roman domination stable upon edge addition, or just $\gamma_R$-EA-stable, if $\gamma_R(G+e)= \gamma_R(G)$ for any edge $e \notin E(G)$. We extend this concept to a weak Roman dominating function as follows: A graph $G$ is said to be weak Roman domination stable upon edge addition, or just $\gamma_r$-EA-stable, if $\gamma_r(G+e)= \gamma_r(G)$ for any edge $e \notin E(G)$. In this paper, we study $\gamma_r$-EA-stable graphs, obtain bounds for &nbsp;$\gamma_r$-EA-stable graphs and&nbsp; characterize $\gamma_r$-EA-stable trees which attain the bound.&nbsp;On the Total Monophonic Number of a Graph
http://comb-opt.azaruniv.ac.ir/article_14419.html
Let G = (V,E) be a connected graph of order n. A path P in G which does not have a chord is called a monophonic path. A subset S of V is called a monophonic set if every vertex v in V lies in a x-y monophonic path where x, y 2 S. If further the induced subgraph G[S] has no isolated vertices, then S is called a total monophonic set. The total monophonic number mt(G) and the upper total monophonic number m+t (G) are respectively the minimum cardinality of a total monophonic set and the maximum cardinality of a minimal total monophonic set. In this paper we determine the value of these parameters for some classes of graphs and establish bounds for the same. We also prove the existence of graphs with prescribed values for mt(G) and m+t (G).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 $Sp(G).$An upper bound on triple Roman domination
http://comb-opt.azaruniv.ac.ir/article_14421.html
For a graph $G=(V,E)$, a triple Roman dominating function (3RD-function) is a function $f:V\to \{0,1,2,3,4\}$ having the property that (i) if $f(v)=0$ then $v$ must have either one neighbor $u$ with $f(u)=4$, or two neighbors $u,w$ with $f(u)+f(w)\ge 5$ or three neighbors $u,w,z$ with $f(u)=f(w)=f(z)=2$, (ii) if $f(v)=1$ then $v$ must have one neighbor $u$ with $f(u)\ge 3$ or two neighbors $u,w$ with $f(u)=f(w)=2$, and (iii) if $f(v)=2$ then $v$ must have one neighbor $u$ with $f(u)\ge 2$. The weight of a 3RDF $f$ is the sum $f(V)=\sum_{v\in V} f(v)$, and the minimum weight of a 3RD-function on $G$ is the triple Roman domination number of $G$, denoted by $\gamma_{[3R]}(G)$. In this paper, we prove that for any connected graph $G$ of order $n$ with minimum degree at least two,&nbsp; $\gamma_{[3R]}(G)\leq \frac{3n}{2}$.On Sombor coindex of graphs
http://comb-opt.azaruniv.ac.ir/article_14422.html
In this paper, we explore several properties of Sombor coindex of a finite simple graph and we derive a bound for the total Sombor index. We also explore its relations to the Sombor index, the Zagreb coindices, forgotten coindex and other important graph parameters. We further compute the bounds of the Somber coindex of some graph operations and derived explicit formulae of Sombor coindex for some well-known graphs as application.Signless Laplacian eigenvalues of the zero divisor graph associated to finite commutative ring $ \mathbb{Z}_{p^{M_{1}}~q^{M_{2}}} $
http://comb-opt.azaruniv.ac.ir/article_14423.html
For a commutative ring $R$ with identity $1\neq 0$, let the set $Z(R)$ denote the set of zero-ivisors and let $Z^{*}(R)=Z(R)\setminus \{0\}$ be the set of non-zero zero-divisors of $R$. &nbsp;The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is a simple graph whose vertex set is $Z^{*} (R)$ and two vertices $u, v \in Z^*(R)$ are adjacent if and only if $uv=vu=0$. In this article, we find the signless Laplacian spectrum of the zero divisor graphs $ \Gamma(\mathbb{Z}_{n}) $ for $ n=p^{M_{1}}q^{M_{2}}$, where $ p&lt;q $ are primes and $ M_{1} , M_{2} $ are positive integers.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.Double Roman domination in graphs: algorithmic complexity
http://comb-opt.azaruniv.ac.ir/article_14425.html
Let $G=(V,E)$ be a graph. &nbsp;A double Roman dominating function &nbsp;(DRDF) of &nbsp; $G $ &nbsp; is a function &nbsp; $f:V\to \{0,1,2,3\}$&nbsp; such that, for each $v\in V$ with $f(v)=0$, &nbsp;there is a vertex $u $ &nbsp;adjacent to $v$ &nbsp;with $f(u)=3$ or there are vertices $x$ and $y $ &nbsp;adjacent to $v$ &nbsp;such that &nbsp;$f(x)=f(y)=2$ and for each $v\in V$ with $f(v)=1$, &nbsp;there is a vertex $u $ &nbsp; &nbsp;adjacent to $v$ &nbsp; &nbsp;with &nbsp;$f(u)&gt;1$. &nbsp;The weight of a DRDF $f$ is &nbsp; $ f (V) =\sum_{ v\in V} f (v)$. &nbsp; Let $n$ and &nbsp;$k$ be integers such that &nbsp;$3\leq 2k+ 1 &nbsp;\leq n$. &nbsp;The &nbsp; generalized Petersen graph $GP (n, k)=(V,E) $ &nbsp;is the &nbsp;graph &nbsp;with &nbsp;$V=\{u_1, u_2,\ldots, &nbsp;u_n\}\cup\{v_1, v_2,\ldots, v_n\}$ and $E=\{u_iu_{i+1}, u_iv_i, v_iv_{i+k}: &nbsp;1 \leq i \leq n\}$, where &nbsp;addition is taken&nbsp; modulo $n$. In this paper, &nbsp;we firstly &nbsp; prove that the &nbsp;decision &nbsp; &nbsp; problem &nbsp;associated with &nbsp; double Roman domination is NP-omplete even restricted to planar bipartite graphs with maximum degree at most 4. &nbsp;Next, we &nbsp; give &nbsp;a dynamic programming algorithm for &nbsp;computing a minimum DRDF (i.e., a &nbsp;DRDF &nbsp; with minimum weight &nbsp;along &nbsp;all &nbsp; DRDFs) &nbsp;of $GP(n,k )$ &nbsp;in $O(n81^k)$ time and space &nbsp;and so a &nbsp;minimum DRDF &nbsp;of $GP(n,O(1))$ &nbsp;can be computed in $O( n)$ time and space.Total restrained Roman domination
http://comb-opt.azaruniv.ac.ir/article_14426.html
Let $G$ be a graph with vertex set $V(G)$. A Roman dominating function &nbsp;(RDF) on a graph $G$ is a function $f:V(G)\longrightarrow\{0,1,2\}$ such that every vertex $v$ &nbsp;with $f(v)=0$ is adjacent to a vertex &nbsp;$u$ with $f(u)=2$. If $f$ is an RDF on $G$, then let $V_i=\{v\in V(G): f(v)=i\}$ for $i\in\{0,1,2\}$. An RDF $f$ is called a restrained (total) Roman dominating function if the subgraph induced by $V_0$ &nbsp;(induced by $V_1\cup V_2$) has no isolated vertex. A total and restrained Roman dominating function is a total restrained Roman &nbsp;dominating function. &nbsp;The total restrained Roman domination number $\gamma_{trR}(G)$ on a graph $G$ is the minimum weight of a total restrained Roman dominating function on the graph $G$. We initiate the study of total restrained Roman domination number and present several sharp bounds on $\gamma_{trR}G)$. In addition, we determine this parameter for some classes of graphs.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.Previous algorithms for constructing a FP are primarily restricted to the cases given below:(i) 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).(ii) A bi-connected plane triangulation with an exterior face of length 3 and no CIPs, known as maximal planar graph (MPG).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 $G.$A homogeneous predictor-corrector algorithm for stochastic nonsymmetric convex conic optimization with discrete support
http://comb-opt.azaruniv.ac.ir/article_14429.html
We consider a stochastic convex optimization problem over nonsymmetric cones with discrete support. This class of optimization problems has not been studied yet. By using a logarithmically homogeneous self-concordant barrier function, we present a homogeneous predictor-corrector interior-point algorithm for solving stochastic nonsymmetric conic optimization problems. We also derive an iteration bound for the proposed algorithm. Our main result is that we uniquely combine a nonsymmetric algorithm with efficient methods for computing the predictor and corrector directions. Finally, we describe a realistic application and present computational results for instances of the stochastic facility location problem formulated as a stochastic nonsymmetric convex conic optimization problem.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.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.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 $\mathcal{EG}(\mathcal{S})$.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.A note on odd facial total-coloring of cacti
http://comb-opt.azaruniv.ac.ir/article_14458.html
A facial total-coloring of a plane graph $G$ is a coloring of the vertices and edges such that no facially adjacent edges, no adjacent vertices, and no edge and its endvertices are assigned the same color. A facial total-coloring of $G$ is odd if for every face $f$ and every color $c$, either no element or an odd number of elements incident with $f$ is colored by $c$. In this note we prove that every cactus forest with an outerplane embedding admits an odd facial total-coloring with at most 16 colors. Moreover, this bound is tight.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.Time-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.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
([23]) proposed an optimization model to deploy and employ unmanned aerial
vehicles unmanned aerial vehicles (UAVs) in special operations missions. Royst et al. ([24])
formulate and solve a routing problem encountered in aircraft mission planning in the presence
of ground threats.
Tian et al. ([25]) solved the problem of routing multiple drones to detect targets with a
time window at a minimal cost. Constraints meet the segregation requirements for identifying
targets and the requirement not to violate the maximum travel time. A GA-based approach is
proposed to solve the problem. Liu et al. ([27]) consider the problem of drone routing as the
task of monitoring traffic for a roadSigned 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.