Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301Maximal outerplanar graphs with semipaired domination number double the domination number1201474610.22049/cco.2024.28972.1801ENMichael A.HenningDepartment of Mathematics and Applied Mathematics, University of Johannesburg,
Auckland Park, 2006 South AfricaPawatonKaemawichanuratMathematics and Statistics with Applications (MaSA)Department of Mathematics, King Mongkut’s University of Technology Thonburi, Bangkok, ThailandJournal Article20230903A subset $S$ of vertices in a graph $G$ is a dominating set if every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. If the graph $G$ has no isolated vertex, then a pair dominating set $S$ of $G$ is a dominating set of $G$ such that $G[S]$ has a perfect matching. Further, a semipaired dominating set of $G$ is a dominating set of $G$ with the additional property that the set $S$ can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The domination number $\gamma(G)$ is the minimum cardinality of a dominating set of $G$. Similarly, the paired (semipaired) domination number $\gamma_{pr}(G)$ $(\gamma_{pr2}(G))$ is the minimum cardinality of a paired (semipaired) dominating set of $G$. It is known that for a graph $G$, $\gamma(G) \le \gamma_{pr2}(G) \le \gamma_{pr}(G) \le 2\gamma(G)$. In this paper, we characterize maximal outerplanar graphs $G$ satisfying $\gamma_{pr2}(G) = 2\gamma(G)$. Hence, our result yields the characterization of maximal outerplanar graphs $G$ satisfying $\gamma_{pr}(G) = 2\gamma(G)$.https://comb-opt.azaruniv.ac.ir/article_14746_fdfdaead6410a7e7e69b1e4a233cf72d.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301γ-Total Dominating Graphs of Lollipop, Umbrella, and Coconut Graphs21321475310.22049/cco.2024.27940.1401ENPannawatEakawinrujeeThammasat Secondary School, Faculty of Learning Sciences and Education,
Thammasat University, Pathum Thani 12121, Thailand0000-0003-1336-1019NantapathTrakultraiprukDepartment of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathum Thani 12120, ThailandJournal Article20220729A total dominating set of a graph $G$ is a set $D\subseteq V(G)$ such that every vertex of $G$ is adjacent to some vertex in $D$. The total domination number $\gamma_{t}(G)$ of $G$ is the minimum cardinality of a total dominating set. The $\gamma$-total dominating graph $TD_{\gamma}(G)$ of $G$ is the graph whose vertices are minimum total dominating sets, and two minimum total dominating sets of $TD_{\gamma}(G)$ are adjacent if they differ by only one vertex. In this paper, we determine the total domination numbers of lollipop graphs, umbrella graphs, and coconut graphs, and especially their $\gamma$-total dominating graphs.https://comb-opt.azaruniv.ac.ir/article_14753_899449515422196bfcaf680b6b0a9120.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301On the comaximal graph of a non-quasi-local atomic domain33481475410.22049/cco.2024.28332.1508ENSubramanianVisweswaranDepartment of Mathematics, Saurashtra University, Rajkot, IndiaPremkumar T.LalchandaniDepartment of Mathematics, Dr. Subhash University, Dr. Subhash Road Junagadh, IndiaJournal Article20230218Let $R$ be an atomic domain such that $R$ has at least two maximal ideals. Let $Irr(R)$ denote the set of all irreducible elements of $R$ and let $J(R)$ denote the Jacobson radical of $R$. Let $\mathcal{I}(R) = \{R\pi\mid \pi\in Irr(R)\backslash J(R)\}$. In this paper, with $R$, we associate an undirected graph denoted by $\mathbb{CGI}(R)$ whose vertex set is $\mathcal{I}(R)$ and distinct vertices $R\pi_{1}$ and $R\pi_{2}$ are adjacent if and only if $R\pi_{1} + R\pi_{2} = R$. The aim of this paper is to study the interplay between some graph properties of $\mathbb{CGI}(R)$ and the algebraic properties of $R$. https://comb-opt.azaruniv.ac.ir/article_14754_cf1e0a5f6303a0f6b61882d095d5f5ce.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301Some properties of star-perfect graphs49561460210.22049/cco.2023.28076.1484ENSanghitaGhoshDepartment of Mathematics, CHRIST(Deemed to be University), Bengaluru, IndiaRavindraGundgurtiDepartment of Mathematics, CHRIST(Deemed to be University), Bengaluru, IndiaAbrahamVettiyankalDepartment of Mathematics, CHRIST(Deemed to be University), Bengaluru, IndiaJournal Article20230107For 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.https://comb-opt.azaruniv.ac.ir/article_14602_14b144a2c140521caa92fa447a2d2195.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301A classification of graphs through quadratic embedding constants and clique graph insights57771476110.22049/cco.2024.29159.1869ENEdy TriBaskoroCombinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences,
Institut Teknologi Bandung, Jalan Ganesa 10 Bandung, IndonesiaNobuakiObataCenter for Data-driven Science and Artificial Intelligence, Tohoku University, Sendai 980-8576, JapanJournal Article20231108The quadratic embedding constant (QEC) of a graph $G$ is a new numeric invariant, which is defined in terms of the distance matrix and is denoted by $\mathrm{QEC}(G)$. By observing graph structure of the maximal cliques <br />(clique graph), we show that a graph $G$ with $\mathrm{QEC}(G)<-1/2$ admits a ``cactus-like'' structure. We derive a formula for the quadratic embedding constant of a graph consisting of two maximal cliques. As an application we discuss characterization of graphs along the increasing sequence of $\mathrm{QEC}(P_d)$, where $P_d$ is the path on $d$ vertices. In particular, we determine graphs $G$ satisfying $\mathrm{QEC}(G)<\mathrm{QEC}(P_5)$.https://comb-opt.azaruniv.ac.ir/article_14761_e0201fc6bfc859df2ca6f4416464ebcc.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-2128111202603012-rainbow domination number of the subdivision of graphs79911477510.22049/cco.2024.28850.1749ENRostamYarke SalkhoriDepartment of Mathematics, Faculty of Science, Imam Khomeini International UniversityEbrahimVatandoostDepartment of Mathematics, Faculty of Science, Imam Khomeini International UniversityAliBehtoeiDepartment of Mathematics, Faculty of Science, Imam Khomeini International UniversityJournal Article20230719Let $G$ be a simple graph and $f : V (G) \rightarrow P(\{1,2\})$ be a function where for each vertex $v \in V (G)$ with $f(v)= \emptyset$ we have $\bigcup_{u \in N_{G}(v)} f(u) = \{1,2\}.$ Then $f$ is a $2$-rainbow dominating function (a $2RDF$) of $G.$ The weight of $f$ is $\omega(f)=\sum_{v \in V(G)} |f(v)|.$ The minimum weight among all of $2-$rainbow dominating functions is $2-$rainbow domination number and is denoted by $\gamma_{r2}(G)$. In this paper, we provide some bounds for the $2-$rainbow domination number of the subdivision graph $S(G)$ of a graph $G$. Also, among some other interesting results, we determine the exact value of $\gamma_{r2}(S(G))$ when $G$ is a tree, a bipartite graph, $K_{r,s}$, $K_{n_1,n_2,\dots,n_k}$ and $K_n$.https://comb-opt.azaruniv.ac.ir/article_14775_86b3bcb4c5c86fa321b24b99c45e48f1.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301Optimization problems with nonconvex multiobjective generalized Nash equilibrium problem constraints931161478110.22049/cco.2024.29435.1993ENEl-yahyaouiYounessLaboratory LASMA, Department of mathematics, Sidi Mohamed Ben Abdellah University,
Fez, MoroccoLahoussineLafhimLaboratory LASMA, Department of mathematics, Sidi Mohamed Ben Abdellah University, Fez, MoroccoJournal Article20240206This work discusses a category of optimization problems in which the lower-level problems include multiobjective generalized Nash equilibrium problems. Despite the fact that it has various possible applications, there has been little research into it in the literature. We provide a single-level reformulation for these types of problems and highlight their equivalence in terms of global and local minimizers. Our method consists of transforming our problem into a one-level optimization problem, utilizing the kth-objective weighted-constraint and optimal value reformulation. The Mordukhovich generalized differentiation calculus is then used to derive completely detailed first-order necessary optimality conditions in the smooth setting.https://comb-opt.azaruniv.ac.ir/article_14781_45062d4c6f712be4f2c6b0649cfd0760.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301Sharp bounds on additively weighted Mostar index of cacti1171301475910.22049/cco.2024.28757.1702ENLijuAlexDepartment of Mathematics,Bishop Chulaparambil Memorial College, Kottayam-686001, IndiaDepartment of Mathematics, Marthoma College, Tiruvalla - 689103, IndiaG.IndulalDepartment of Mathematics, St. Aloysius College, Edathua, Alappuzha - 689573, IndiaJournal Article20230616Let C(n, t) denotes the collection of all cacti of order n with exactly t cycles and Ctn<br /><br />denotes the cacti of order n and t end vertices. In this paper, we compute the upper bound, second largest upper bound, and third largest upper bound of the additively weighted Mostar index of graphs in C(n, t). We also determine the upper bound of the additively weighted Mostar index for cacti of order n with a fixed number of end vertices. We characterize all the graphs attaining the bounds.https://comb-opt.azaruniv.ac.ir/article_14759_d9e7bd89803f5cc3db7b689b00e8c8f1.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301Hypergraphs defined on algebraic structures1311441476010.22049/cco.2024.29607.2077ENPeter J.CameronSchool of Mathematics and Statistics
University of St. Andrews
Fife, UKAparna LakshmananSDepartment of Mathematics
Cochin University of Science and Technology, Cochin - 22
Kerala, IndiaMidhuna V.AjithDepartment of Mathematics
Cochin University of Science and Technology, Cochin - 22
Kerala, IndiaJournal Article20240325There has been a great deal of research on graphs defined on algebraic structures in the last two decades. Power graphs, commuting graphs, cyclic graphs are some examples. In this paper we begin an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this can add a new perspective.https://comb-opt.azaruniv.ac.ir/article_14760_3009abd77d89af8ebbbd3ad2fd730517.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301Strength based domination in graphs1451541478210.22049/cco.2024.29456.2002ENA.LekhaDepartment of Mathematics,
Government Engineering College},
Thrissur-680 009, Kerala, IndiaK.S.ParvathyDepartment of Mathematics,
St. Mary's College,
Thrissur-680 020, Kerala, IndiaS.ArumugamAdjunct Professor, Department of Computer Science and Engineering,
Ramco Institute of Technology, Rajapalayam-626117, Tamilnadu, IndiaJournal Article20240216Let $G=(V,E)$ be a connected graph. Let $A\subseteq V$ and $v\in V-A.$ The dominating strength of $A$ on $v$ is defined by $s(v,A)=\sum\limits_{u\in A}\frac{1}{d(u,v)}.$ A subset $D$ of $V$ is called a strength based dominating set if for every vertex $v\notin D,$ there exists a subset $A$ of $D$ such that $s(v,A)\geq 1.$ The $sb$-domination number $\gamma_{sb}(G)$ is the minimum cardinality of a strength based dominating set of $G.$ In this paper we initiate a study of this parameter and indicate directions for further research.https://comb-opt.azaruniv.ac.ir/article_14782_459cf166035d78b84de37443c0ac9cb2.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301On the $A_{\alpha}$-spectrum of the $k$-splitting signed graph and neighbourhood coronas1551691479610.22049/cco.2024.29723.2133ENShariefuddinPirzadaDepartment of Mathematics, University of Kashmir, Srinagar, IndiaMir Riyaz UlRashidDepartment of Mathematics, University of Kashmir, Srinagar, IndiaJournal Article20240510Let $\Sigma=(G,\sigma)$ be a signed graph with adjacency matrix $A(\Sigma)$ and $D(G)$ be the diagonal matrix of its vertex degrees. For any real $\alpha\in [0,1]$, the $A_{\alpha}$-matrix of a signed graph $\Sigma$ is defined as $A_{\alpha}(\Sigma)=\alpha D(G)+(1-\alpha)A(\Sigma)$. Given a signed graph $\Sigma$ with vertex set $V=\{v_1, v_2,\dots, v_n\}$, the $k$-splitting signed graph $SP_k(\Sigma)$ of $\Sigma$ is obtained by adding to each vertex $v\in V(\Sigma)$ new $k$ vertices say $u^1, u^2, \ldots, u^k$ and joining every neighbour say $u$ of the vertex $v$ to $u^i$, $1\le i\le k$ by an edge which inherits the sign from $uv$. In this paper, we determine the $A_{\alpha}$-spectrum of $SP_k(\Sigma)$ in case of $\Sigma$ being a regular signed graph. For $k=1$, we introduce two distinct coronas of signed graphs $\Sigma_1$ and $\Sigma_2$ based on $SP_1(\Sigma_1)$, namely the splitting V-vertex neighbourhood corona and the splitting S-vertex neighbourhood corona. By examining the $A_{\alpha}$-characteristic polynomial of the resulting signed graphs, we derive their $A_{\alpha}$-spectra under certain regularity conditions on the constituent signed graphs. As applications, we use these results to construct infinite pairs of nonregular $A_{\alpha}$-cospectral signed graphs.https://comb-opt.azaruniv.ac.ir/article_14796_25dec0f2ac937dda7bb0167f559f44c7.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301Two techniques to reduce the Pareto optimal solutions in multiobjective optimization problems1711881478310.22049/cco.2024.28753.1700ENFatemehAhmadiDepartment of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, IranDavoudForoutanniaDepartment of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, IranJournal Article20230615In this study, for a decomposed multi-objective optimization problem, we propose the direct sum of the preference matrices of the subproblems provided by the decision maker (DM). Then, using this matrix, we present a new generalization of the rational efficiency concept for solving the multi-objective optimization problem (MOP). A problem that sometimes occurs in multi-objective optimization is the existence of a large set of Pareto optimal solutions. Hence, decision making based on selecting a unique preferred solution becomes difficult. Considering models with the concept of generalized rational efficiency can relieve some of the burden from the DM by shrinking the solution set. This paper discusses both theoretical and practical aspects of rationally efficient solutions related to this concept. Moreover, we present two techniques to reduce the Pareto optimal solutions using. The first technique involves using the powers of the preference matrix, while the second technique involves creating a new preference matrix by modifying the decomposition of the MOP.https://comb-opt.azaruniv.ac.ir/article_14783_dff75bc2316f0d3a132ccb79852c6a9c.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301Edge corona product and its topological descriptors with applications in complex molecular structures1892031478610.22049/cco.2024.29326.1943ENMuhammad FaisalBashirDepartment of Mathematics, Riphah International University, Lahore, PakistanMuhammad KamranJamilDepartment of Mathematics, Riphah International University, Lahore, PakistanMuhammadWaheedPunjab School Education Department, Govt Higher Secondary School Nangal Sahdan,
Muridke 39000, Sheikhupura, PakistanAishaJavedAbdus Salam School of Mathematical Sciences, Government College University, Lahore, PakistanIsmailNaci CangulFaculty of Arts and Science, Department of Mathematics, Bursa Uludag University, Gorukle 16059 Bursa, TurkeyJournal Article20240102Graph operations offer a robust framework that enables the analysis, modeling, and resolution of intricate problems. Their versatility and broad range of applications make them essential across numerous fields of study and research, playing an irreplaceable role in tackling complex challenges. A topological index is a real number associated with a graph that gives insight into the topological properties of the graph. There are numerous topological indices in this era now, with three variants like degree based, distance based and eccentricity based topological indices. In this paper, we studied a well known graph operation named as edge corona product and investigate their some degree based topological indices. As applications, this graph operations can be used to study topological properties of complex structure of linear and cyclic silicate networks, together with triangular and double triangular networks. Some existing results in the literature can be obtained as corollaries of the new results. A conjecture is proposed relating the general first Zagreb index of the edge corona product of two graphs.https://comb-opt.azaruniv.ac.ir/article_14786_738e23b98448395590fb4739bae2242f.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301A note on graphs with integer Sombor index2052091478910.22049/cco.2024.29474.2013ENNoor A'lawiahAbd AzizSchool of Mathematical Sciences , Universiti Sains Malaysia, 11800 Penang, MalaysiaJournal Article20240221For a graph $G$, the Sombor index of $G$ is defined as $ SO(G)=\sum_{uv\in E(G)} \sqrt{\deg(u)^2+\deg(v)^2}$, where $\deg(u)$ is referring to the degree of vertex $u$ in $G$. In this paper, we present a construction, namely $R_k$-construction which produce infinitely many families of graphs whose Sombor indices are integers.https://comb-opt.azaruniv.ac.ir/article_14789_a6e973a7b21b077a2109c33700442107.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301Remarks on the bounds of graph energy2112261479510.22049/cco.2024.28825.1739ENŞ. BurcuBozkurt AltındağDepartment of Mathematics, Faculty of Science Sel¸cuk University, Konya, TurkeyEminaMilovanovićUniversity of Niš, Faculty of Electronic Engineering, Niš, SerbiaMarjanMatejićUniversity of Niš, Faculty of Electronic Engineering, Niš, SerbiaIgorMilovanovićUniversity of Niš, Faculty of Electronic Engineering, Niš, SerbiaJournal Article20230713Let $G$ be a graph of order $n$ with eigenvalues $\lambda _{1}\geq \lambda_{2}\geq \cdots \geq \lambda _{n}.$ The energy of $G$ is defined as $E\left(G\right) =\sum_{i=1}^{n}\left\vert \lambda _{i}\right\vert $. In the present paper, new bounds on $E(G)$ are provided. In addition, some bounds of $E(G)$ are compared. https://comb-opt.azaruniv.ac.ir/article_14795_ae6efd834babceea1fb92d6bcf6197b8.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301On cozero divisor graphs of ring $Z_n$2272451480610.22049/cco.2024.26112.1974ENZahidRazaDepartment of Mathematics, College of Sciences, University of Sharjah, UAEBilal AhmadRatherMathematical Sciences Department, College of Science, United Arab Emirates University, Al Ain
15551, Abu Dhabi, UAEModjtabaGhorbaniDepartment of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University,
Tehran, 16785-163, I.R. Iran0000-0001-5623-9932Journal Article20240124The cozero divisor graph $\Gamma^{\prime}(R)$ of a commutative ring $R$ is a simple graph with vertex set as non-zero zero divisor elements of $R$ such that two distinct vertices $x$ and $y$ are adjacent iff $x\notin Ry$ and $y\notin Rx$, where $xR$ is the ideal generated by $x$. In this article we find the spectra of $\Gamma^{\prime}(\mathbb{Z}_{n}) $ for $n\in \{q_{1}q_{2}, q_{1}q_{2}q_{3},q_{1}^{n_{1}}q_{2}\},$ where $q_{i}$'s are primes. As a consequence we obtain the bounds for the largest (smallest) eigenvalues, bounds for spread, rank and inertia of $ \Gamma^{\prime}(\mathbb{Z}_{q_{1}^{n_{1}}q_{2}})$ along with the determinant, inverse and square of trace of its quotient matrix. We present the extremal bounds for the energy of $\Gamma^{\prime}(\mathbb{Z}_{n})$ for $n=q_{1}^{n_{1}}q_{2}$ and characterize the extremal graphs attaining them. We close article with conclusion for furtherance.https://comb-opt.azaruniv.ac.ir/article_14806_703650707f47137f3e80cbff75f87269.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301Total double Roman domination stability in graphs2472581496610.22049/cco.2025.30402.2453ENZiqiangXuInstitute of Computing Science and Technology, Guangzhou University,
Guangzhou 510006, ChinaSaeedKosariInstitute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China0000-0002-1427-5473MinaEsmaeiliDepartment of Mathematics,
Azarbaijan Shahid Madani University,
Tabriz, I.R. IranAyshaKhanUniversity of Technology and Applied Sciences, Musannah,
OmanLutzVolkmannRWTH Aachen,
52056 Aachen, GermanyJournal Article20250307Let $G$ be a graph with vertex set $V(G)$. A total double Roman dominating function (TDRD-function) on a graph $G$ with no isolated vertices is a function $f :V(G)\to \{0, 1, 2, 3\}$ satisfying the conditions: $(i)$ if $f(v)=0$, then the vertex $v$ must be adjacent to at least two vertices assigned $2$ or one vertex assigned $3$ under $f$, and if $f(v)=1$, then the vertex $v$ must be adjacent to at least one vertex assigned $2$ or $3$ and $(ii)$ the subgraph of $G$ induced by the set $\{v \in V(G) \mid f(v)\neq 0\}$ has no isolated vertices. The weight of a TDRD-function $f$ is the sum of its function values over all vertices, and the minimum weight of a TDRD-function on $G$ is the total double Roman domination number, $\gamma_{tdR}(G)$. The $\gamma_{tdR}$-stability ($\gamma^-_{tdR}$-stability, $\gamma^+_{tdR}$-stability) of $G$, denoted by ${\rm st}_{\gamma_{tdR}}(G)$ (resp. ${\rm st}^-_{\gamma_{tdR}}(G)$, ${\rm st}^+_{\gamma_{tdR}}(G)$), is defined as the minimum size of a set of vertices whose removal changes (resp. decreases, increases) the total double Roman domination number. In this paper, we first determine the exact values of the $\gamma_{tdR}$-stability of some special classes of graphs, and then we present some bounds on ${\rm st}_{\gamma_{tdR}}(G)$, ${\rm st}^-{\gamma_{tdR}}(G)$ and ${\rm st}^+_{\gamma_{tdR}}(G)$). In particular, for a graph $G$ with maximum degree $\Delta\ge 3$, we show that ${\rm st}^-_{\gamma_{tdR}}(G)\leq \Delta-1$.https://comb-opt.azaruniv.ac.ir/article_14966_3bc87007f47e706e48f1b9897316823f.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301Irredundance chromatic number and gamma chromatic number of trees2692751482510.22049/cco.2024.29328.1945ENDavidA KalarkopDepartment of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, ThailandPawatonKaemawichanuratDepartment of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, ThailandMathematics and Statistics with Application (MaSA)Journal Article20240103A vertex subset $S$ of a graph $G = (V, E)$ is irredundant if every vertex in $S$ has a private neighbor with respect to $S$. An irredundant set $S$ of $G$ is maximal if, for any $v \in V - S$, the set $S \cup \{v\}$ is no longer irredundant. The lower irredundance number of $G$ is the minimum cardinality of a maximal irredundant set of $G$ and is denoted by $ir(G)$. A coloring $\mathcal{C}$ of $G$ is said to be the irredundance coloring if there exists a maximal irredundant set $R$ of $G$ such that all the vertices of $R$ receive different colors. The minimum number of colors required for an irredundance coloring of $G$ is called the irredundance chromatic number of $G$, and is denoted by $\chi_{i}(G)$. A coloring $\mathcal{C}$ of $G$ is said to be the gamma coloring if there exists a dominating set $D$ of $G$ such that all the vertices of $D$ receive different colors. The minimum number of colors required for a gamma coloring of $G$ is called the gamma chromatic number of $G$, and is denoted by $\chi_{\gamma}(G)$. In this paper, we prove that every tree $T$ satisfies $\chi_{i}(T) = ir(T)$ unless $T$ is a star. Further, we prove that $\gamma(T) \leq \chi_{\gamma}(T) \leq \gamma(T) + 1$. We characterize all trees satisfying the upper bound.https://comb-opt.azaruniv.ac.ir/article_14825_eb0122e36f6ffd669ef915f0fa485d08.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301On the reciprocal distance Laplacian spectral radius of graphs2592681480310.22049/cco.2024.29493.2024ENUmmerMushtaqDepartment of Mathematics, University of Kashmir, Srinagar, IndiaShariefuddinPirzadaDepartment of Mathematics, University of Kashmir, Srinagar, IndiaJournal Article20240301The reciprocal distance Laplacian matrix of a connected graph $G$ is defined as $RD^L(G)=RTr(G)-RD(G)$, where $RTr(G)$ is the diagonal matrix whose $i$-th element $RTr(v_i)=\sum_{i\ne j\in V(G)} \frac{1}{d_{ij}}$ and $RD(G)$ is the Harary matrix. $RD^L(G)$ is a real symmetric matrix and we denote its eigenvalues as $\lambda_1(RD^L(G))\geq \lambda_2(RD^L(G))\geq\dots\geq\lambda_n(RD^L(G))$. The largest eigenvalue $\lambda_1(RD^L(G))$ of $RD^L(G)$ is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain upper bounds for the reciprocal distance Laplacian spectral radius. We characterize the extremal graphs attaining this bound.https://comb-opt.azaruniv.ac.ir/article_14803_fff8c978a75649132c0687ec026d0eca.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301New bounds on distance Estrada index of graphs2772851478510.22049/cco.2024.29488.2022ENMohammad RezaOboudiDepartment of Mathematics, College of Sciences, Shiraz University, Shiraz, 71457-44776, IranJournal Article20240229For a connected graph $G$ with vertex set $\{v_1,\ldots,v_n\}$, the distance matrix of $G$, denoted by $D(G)$, is an $n\times n$ matrix with zero main diagonal, such that its $(i,j)$-entry is $d(v_i,v_j)$, where $i\neq j$ and $d(v_i,v_j)$ is the distance between $v_i$ and $v_j$. Let $\theta_1,\ldots,\theta_n$ be the eigenvalues of $D(G)$. The distance Estrada index of $G$ is defined as $DEE(G)=\sum_{i=1}^ne^{\theta_i}$. In this paper we find some new sharp bounds for the distance Estrada index of graphs. Our results improve the previous bounds on the distance Estrada index of graphs.https://comb-opt.azaruniv.ac.ir/article_14785_5e38fef0d496b50acbcda61fd1302e22.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301Edge metric dimension of silicate networks2872961480810.22049/cco.2024.29553.2052ENSavariPrabhuDepartment of Mathematics, Rajalakshmi Engineering College, Chennai 602105, IndiaTJenifer JananyDepartment of Mathematics, Rajalakshmi Engineering College, Chennai 602105, India0000-0002-8368-6882Journal Article20240318Metric dimension is an essential parameter in graph theory that aids in addressing issues pertaining to information retrieval, localization, network design, and chemistry through the identification of the least possible number of elements necessary to identify the vertices in a graph uniquely. A variant of metric dimension, called the edge metric dimension focuses on distinguishing the edges in a graph $G$, with a vertex subset. The minimum possible number of vertices in such a set is denoted as $\dim_E(G)$. This paper presents the precise edge metric dimension of silicate networks.https://comb-opt.azaruniv.ac.ir/article_14808_0b17b3a84e11cdeece0a76cee2362065.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811120260301On the energy of the line graph of unitary Cayley graphs2973061480410.22049/cco.2024.29624.2091ENFatemeMovahediDepartment of Mathematics, Faculty of Sciences, Golestan University, Gorgan, IranJournal Article20240401The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of the line graph of graph $G$ is denoted by $E(L(G))$. The unitary Cayley graph $X_n$ is a graph with the vertex set $Z_n=\{0, 1, \ldots, n-1\}$ and the edge set $\{(a,b) \, : \, ged(a-b,n)=1\}$. In this paper, we focus on the line graph of the unitary Cayley graph $X_n$ and compute the spectrum of line graphs of $X_n$ and its complement graph $\overline{X_n}$. We also obtain the energy of the line graph of $X_n$ and $\overline{X_n}$.https://comb-opt.azaruniv.ac.ir/article_14804_8145e7223dc40a567cfbb3deffd4f1c7.pdf