University of Isfahan Transactions on Combinatorics 2251-8657 1 4 2012 12 01 Hosoya and Merrifield-Simmons indices of some classes of corona of two graphs 1 7 1946 10.22108/toc.2012.1946 EN Mohammad Hossein Reyhani Islamic Azad University, Yazd branch Saeid Alikhani Yazd University Mohammad Ali Iranmanesh Yazd University Journal Article 2012 06 12 Let \$G=(V,E)\$ be a‎ ‎simple graph of order \$n\$ and size \$m\$‎. ‎An \$r\$-matching of \$G\$ is‎ ‎a set of \$r\$ edges of \$G\$ which no two of them have common vertex‎. ‎The Hosoya index \$Z(G)\$ of a graph \$G\$ is defined as the total‎ ‎number of its matchings‎. ‎An independent set of \$G\$ is a set of‎ ‎vertices where no two vertices are adjacent‎. ‎The‎ ‎Merrifield-Simmons index of \$G\$ is defined as the total number of‎ ‎the independent sets of \$G\$‎. ‎In this paper we obtain Hosoya and‎ ‎Merrifield-Simmons indices of corona of some graphs‎. http://toc.ui.ac.ir/article_1946_94db2778225c932d0361b35eb33357ba.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 1 4 2012 12 01 Determinants of adjacency matrices of graphs 9 16 2041 10.22108/toc.2012.2041 EN Alireza Abdollahi University of Isfahan Journal Article 2012 06 16 ‎We study the set of all determinants of adjacency matrices of graphs with a given number of vertices‎. ‎Using Brendan McKay's data base of small graphs‎, ‎determinants of graphs with at most \$9\$ vertices are computed so‎ ‎that the number of non-isomorphic graphs with given vertices whose determinants are all equal to a number is exhibited in a table‎. ‎Using an idea of M‎. ‎Newman‎, ‎it is proved that if \$G\$ is a graph with \$n\$ vertices‎, ‎\$m\$ edges and \${d_1,dots,d_n}\$ is the set of vertex degrees of \$G\$‎, ‎then‎ ‎\$gcd(2m,d^2)\$ divides the determinant of the adjacency matrix of \$G\$‎, ‎where \$d=gcd(d_1,dots,d_n)\$‎. ‎Possible determinants of adjacency matrices of graphs with exactly two cycles are obtained‎. http://toc.ui.ac.ir/article_2041_b9579dd3348af7ab9be7b60997202498.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 1 4 2012 12 01 A simple approach to order the multiplicative Zagreb indices of connected graphs 17 24 2146 10.22108/toc.2012.2146 EN Mehdi Eliasi Department of Mathematics and Computer Science , Faculty of Khansar, Khansar, Iran Journal Article 2012 10 30 The first (\$Pi_1\$) and the second \$(Pi_2\$) multiplicative Zagreb indices of a connected graph \$G\$‎, ‎with vertex set‎ ‎\$V(G)\$ and edge set \$E(G)\$‎, ‎are defined as \$Pi_1(G) = prod_{u in‎ ‎V(G)} {d_u}^2\$ and \$Pi_2(G) = prod_{uv in‎‎E(G)} {d_u}d_{v}\$‎, ‎respectively‎, ‎where \${d_u}\$ denotes the degree of the vertex \$u\$‎. ‎In this paper we present a simple approach to order these indices for connected graphs on the same number of vertices‎. ‎Moreover‎, ‎as an application of this‎ ‎simple approach‎, ‎we extend the known ordering of the first and the second multiplicative Zagreb indices‎ ‎for some classes of connected graphs‎. http://toc.ui.ac.ir/article_2146_61a92ef69e13895fcd946bfe86a93cd7.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 1 4 2012 12 01 On label graphoidal covering number-I 25 33 2271 10.22108/toc.2012.2271 EN Ismail Sahul Hamid DEPARTMENT OF MATHEMATICS THE MADURA COLLEGE MADURAI, TAMIL NADU Arumugaperumal Anitha Department of Mathematics Thiagarajar Engineering College Madurai Journal Article 2012 08 06 Let \$G=(V‎, ‎E)\$ be a graph with \$p\$ vertices and \$q\$ edges‎. ‎An <em>acyclic‎ ‎graphoidal cover</em> of \$G\$ is a collection \$psi\$ of paths in \$G\$‎ ‎which are internally-disjoint and cover each edge of the graph‎ ‎exactly once‎. ‎Let \$f‎: ‎Vrightarrow {1‎, ‎2‎, ‎ldots‎, ‎p}\$ be a bijective‎ ‎labeling of the vertices of \$G\$‎. ‎Let \$uparrow!G_f\$ be the‎ ‎directed graph obtained by orienting the edges \$uv\$ of \$G\$ from‎ ‎\$u\$ to \$v\$ provided \$f(u)< f(v)\$‎. ‎If the set \$psi_f\$ of all‎ ‎maximal directed paths in \$uparrow!G_f\$‎, ‎with directions‎ ‎ignored‎, ‎is an acyclic graphoidal cover of \$G\$‎, ‎then \$f\$ is called‎ ‎a emph{graphoidal labeling} of \$G\$ and \$G\$ is called a‎ ‎<em>label graphoidal graph</em> and \$eta_l=min{|psi_f|‎: ‎f {rm ‎is a graphoidal labeling of} G}\$ is called the <em>label‎ ‎graphoidal covering number</em> of \$G\$‎. ‎In this paper we characterize‎ ‎graphs for which (i) \$eta_l=q-m\$‎, ‎where \$m\$ is the number of‎ ‎vertices of degree 2 and (ii) \$eta_l= q\$‎. ‎Also‎, ‎we determine the value of label graphoidal covering number for unicyclic graphs‎. http://toc.ui.ac.ir/article_2271_ac97bba24799ba01acbe600e2d71d194.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 1 4 2012 12 01 Toeplitz graph decomposition 35 41 2168 10.22108/toc.2012.2168 EN Samira Hossein Ghorban PhD Student Journal Article 2012 08 28 ‎Let \$n,,t_1,,ldots,,t_k\$ be distinct positive integers‎. ‎A Toeplitz graph \$G=(V‎, ‎E)\$ is a graph with \$V ={1,ldots,n}\$ and‎ ‎\$E= {(i,j)mid |i-j|in {t_1,ldots,t_k}}\$‎. ‎In this paper‎, ‎we present some results on decomposition of Toeplitz graphs‎. http://toc.ui.ac.ir/article_2168_7210c55e7fbc299f5f866aad7d1281a5.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 1 4 2012 12 01 On a relation between Szeged and Wiener indices of bipartite graphs 43 49 2450 10.22108/toc.2012.2450 EN Lilly Chen Nankai University, Center for Combinatorics Xueliang Li Nankai University Mengmeng Liu Nankai University, Center for Combinatorics Ivan Gutman University of Kragujevac Kragujevac, Serbia Journal Article 2013 01 10 Hansen et‎. ‎al.‎, ‎using the AutoGraphiX software‎ ‎package‎, ‎conjectured that the Szeged index \$Sz(G)\$ and the‎ ‎Wiener index \$W(G)\$ of a connected bipartite graph \$G\$ with \$n geq‎ ‎4\$ vertices and \$m geq n\$ edges‎, ‎obeys the relation‎ ‎\$Sz(G)-W(G) geq 4n-8\$‎. ‎Moreover‎, ‎this bound would be the best possible‎. ‎This paper offers a proof to this conjecture‎. http://toc.ui.ac.ir/article_2450_71f471bc2fe6a994e640ca1d23353d16.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 1 4 2012 12 01 The Hosoya index and the Merrifield-Simmons index of some graphs 51 60 2588 10.22108/toc.2012.2588 EN Asma Hamzeh Tarbiat Modares University Ali Iranmanesh Department of Mathematics, Tarbiat Modares University, P. O. Box 14115-137, Tehran Mohammad Ali Hosseinzadeh Tarbiat Modares University Samaneh Hossein-Zadeh Tarbiat Modares University Journal Article 2012 12 04 ‎The Hosoya index and the Merrifield-Simmons index are two types of‎ ‎graph invariants used in mathematical chemistry‎. ‎In this paper‎, ‎we give some formulas to compute these indices for some classes‎ ‎of corona product and link of two graphs‎. ‎Furthermore‎, ‎we obtain‎ ‎exact formulas of Hosoya and Merrifield-Simmons indices for the‎ ‎set of bicyclic graphs‎, ‎caterpillars and dual star‎. http://toc.ui.ac.ir/article_2588_73c8b27f7c81747f22d33f7d5b65c9ba.pdf