University of IsfahanTransactions on Combinatorics2251-86574420151201Chromatic and clique numbers of a class of perfect graphs14734010.22108/toc.2015.7340ENMohammad RezaFanderAzad University, Chaluse BranchJournal Article20141117Let $p$ 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 $u\neq v$ and $p\nmid u+v$. 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.pdfUniversity of IsfahanTransactions on Combinatorics2251-86574420151201On the harmonic index of graph operations514738910.22108/toc.2015.7389ENB.Shwetha ShettyDon Bosco Institute of Technology, Bangalore-78, IndiaV.LokeshaDept.of Mathematics, VSK University,Bellary
KarnatakaP. S.RanjiniDon Bosco Institute of Technology, Bangalore-78, IndiaJournal Article20141029The harmonic index of a connected graph $G$, denoted by $H(G)$, is defined as $H(G)=\sum_{uv\in E(G)}\frac{2}{d_u+d_v}$ 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.pdfUniversity of IsfahanTransactions on Combinatorics2251-86574420151201A dynamic domination problem in trees1531759010.22108/toc.2015.7590ENWilliamKlostermeyerSchool of Computing
University of North FloridaChristinaMynhardtDepartment of Mathematics and Statistics
University of VictoriaJournal Article20140419We 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.pdfUniversity of IsfahanTransactions on Combinatorics2251-86574420151201The resistance distance and the Kirchhoff index of the $k$-th semi total point graphs3341776710.22108/toc.2015.7767ENDenglanCuiDepartment of Mathematics
Hunan Nornal University
Changsha, Hunan 410081YaopingHouDepartment of Mathematics
Hunan Normal University
Changsha, Hunan,410081Journal Article20141216The $k$-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.pdfUniversity of IsfahanTransactions on Combinatorics2251-86574420151201Broadcast domination in Tori4353765410.22108/toc.2015.7654ENKian WeeSohDept of Mathematics, National University of SingaporeKhee-MengKohDepartment of Mathematics
National University of SingaporeJournal Article20141021A <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.pdfUniversity of IsfahanTransactions on Combinatorics2251-86574420151201A classification of finite groups with integral bi-Cayley graphs5561780710.22108/toc.2015.7807ENMajidArezoomandDepartmant of Mathematical Sciences, Isfahan University of Technology, Isfahan, IranBijanTaeriDepartment of Mathematics, Isfahan University of Technology, Isfahan, IranJournal Article20140714The bi-Cayley graph of a finite group $G$ with respect to a subset $S\subseteq G$, which is denoted by $BCay(G,S)$, is the graph with vertex set $G\times\{1,2\}$ and edge set $\{\{(x,1), (sx,2)\}\mid x\in G, \ s\in S\}$. 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