University of Isfahan Transactions on Combinatorics 2251-8657 4 4 2015 12 01 Chromatic and clique numbers of a class of perfect graphs 1 4 7340 10.22108/toc.2015.7340 EN Mohammad Reza Fander Azad University, Chaluse Branch Journal Article 2014 11 17 ‎Let \$p\$ be a prime number and \$n\$ be a positive integer‎. ‎The graph‎ ‎\$G_p(n)\$ is a graph with vertex set \$[n]={1‎, ‎2,ldots‎, ‎n}\$‎, ‎in‎ ‎which there is an arc from \$u\$ to \$v\$ if and only if \$uneq v\$ and‎ ‎\$pnmid u+v\$‎. ‎In this paper it is shown that \$G_p(n)\$ is a perfect‎ ‎graph‎. ‎In addition‎, ‎an explicit formula for the chromatic number of‎ ‎such graph is given‎. https://toc.ui.ac.ir/article_7340_88e0ca90702ff396dfd3dfe5d04f5bcc.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 4 4 2015 12 01 On the harmonic index of graph operations 5 14 7389 10.22108/toc.2015.7389 EN B. Shwetha Shetty Don Bosco Institute of Technology, Bangalore-78, India V. Lokesha Dept.of Mathematics, VSK University,Bellary Karnataka 0000-0003-2468-9511 P. S. Ranjini Don Bosco Institute of Technology, Bangalore-78, India Journal Article 2014 10 29 ‎‎The harmonic index of a connected graph \$G\$‎, ‎denoted by \$H(G)\$‎, ‎is‎ ‎defined as \$H(G)=sum_{uvin E(G)}frac{2}{d_u+d_v}\$‎ ‎where \$d_v\$ is the degree of a vertex \$v\$ in G‎. ‎In this paper‎, ‎expressions for the Harary indices of the‎ ‎join‎, ‎corona product‎, ‎Cartesian product‎, ‎composition and symmetric difference of graphs are‎ ‎derived‎. <br />  https://toc.ui.ac.ir/article_7389_3e9ec295be34a42ee71a0570cc2fbfa9.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 4 4 2015 12 01 A dynamic domination problem in trees 15 31 7590 10.22108/toc.2015.7590 EN William Klostermeyer School of Computing University of North Florida Christina Mynhardt Department of Mathematics and Statistics University of Victoria 0000-0001-6981-676X Journal Article 2014 04 19 ‎We consider a dynamic domination problem for graphs in which an infinite‎ ‎sequence of attacks occur at vertices with guards and the guard at the‎ ‎attacked vertex is required to vacate the vertex by moving to a neighboring‎ ‎vertex with no guard‎. ‎Other guards are allowed to move at the same time‎, ‎and‎ ‎before and after each attack and the resulting guard movements‎, ‎the vertices‎ ‎containing guards form a dominating set of the graph‎. ‎The minimum number of‎ ‎guards that can successfully defend the graph against such an arbitrary‎ ‎sequence of attacks is the m-eviction number‎. ‎This parameter lies between the‎ ‎domination and independence numbers of the graph‎. ‎We characterize the classes of trees for which the m-eviction number equals‎ ‎the domination number and the independence number‎, ‎respectively‎. <br /><br /> https://toc.ui.ac.ir/article_7590_fbf0dbf66e3b8321a9266cd46dabc47a.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 4 4 2015 12 01 The resistance distance and the Kirchhoff index of the \$k\$-th semi total point graphs 33 41 7767 10.22108/toc.2015.7767 EN Denglan Cui Department of Mathematics Hunan Nornal University Changsha, Hunan 410081 Yaoping Hou Department of Mathematics Hunan Normal University Changsha, Hunan,410081 Journal Article 2014 12 16 ‎The \$k\$-th semi-total point graph \$R^k(G)\$ of a graph \$G\$‎, ‎is a graph obtained from \$G\$ by adding \$k\$ vertices corresponding to each edge and connecting them to the endpoints of the edge considered‎. ‎In this paper‎, ‎we obtain formulas for the resistance distance and Kirchhoff index of \$R^k(G).\$‎   https://toc.ui.ac.ir/article_7767_d003df08d9fc1d3ea3e0f98ad110cc1b.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 4 4 2015 12 01 Broadcast domination in Tori 43 53 7654 10.22108/toc.2015.7654 EN Kian Wee Soh Dept of Mathematics, National University of Singapore Khee-Meng Koh Department of Mathematics National University of Singapore Journal Article 2014 10 21 A <em>broadcast</em> on a graph \$G\$ is a function \$f‎ : ‎V(G) rightarrow {0‎, ‎1,dots‎, ‎diam(G)}\$ such that for every vertex \$v in V(G)\$‎, ‎\$f(v) leq e(v)\$‎, ‎where \$diam(G)\$ is the diameter of \$G\$‎, ‎and \$e(v)\$ is the eccentricity of \$v\$‎. ‎In addition‎, ‎if every vertex hears the broadcast‎, ‎then the broadcast is a <em>dominating broadcast</em>. ‎The <em>cost</em> of a broadcast \$f\$ is the value \$sigma(f) = sum_{v in V(G)} f(v)\$‎. ‎In this paper we determine the minimum cost of a dominating broadcast (also known as the <em>broadcast domination number</em>) for a torus \$C_{m} ;Box; C_{n}\$‎. https://toc.ui.ac.ir/article_7654_d69d40ef7aab331d7b142121c88012e9.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 4 4 2015 12 01 A classification of finite groups with integral bi-Cayley graphs 55 61 7807 10.22108/toc.2015.7807 EN Majid Arezoomand Departmant of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran Bijan Taeri Department of Mathematics, Isfahan University of Technology, Isfahan, Iran Journal Article 2014 07 14 The bi-Cayley graph of a finite group \$G\$ with respect to a subset \$Ssubseteq G\$‎, ‎which is denoted by \$BCay(G,S)\$‎, ‎is the graph with‎ ‎vertex set \$Gtimes{1,2}\$ and edge set \${{(x,1)‎, ‎(sx,2)}mid xin G‎, ‎ sin S}\$‎. ‎A‎ ‎finite group \$G\$ is called a textit{bi-Cayley integral group} if for any subset \$S\$ of‎ ‎\$G\$‎, ‎\$BCay(G,S)\$ is a graph with integer eigenvalues‎. ‎In this paper we prove‎ ‎that a finite group \$G\$ is a bi-Cayley integral group if and only if \$G\$ is isomorphic to‎ ‎one of the groups \$Bbb Z_2^k\$‎, ‎for some \$k\$‎, ‎\$Bbb Z_3\$ or \$S_3\$‎. https://toc.ui.ac.ir/article_7807_741eee48891d7eafd3a189c9e3afd5fb.pdf