University of Isfahan Transactions on Combinatorics 2251-8657 3 4 2014 12 01 Randic incidence energy of graphs 1 9 5573 10.22108/toc.2014.5573 EN Ran Gu Nankai University Fei Huang Nankai University Xueliang Li Center for Combinatorics, Nankai University, Tianjin 300071, China Journal Article 2014 05 29 Let \$G\$ be a simple graph with vertex set \$V(G) = {v_1, v_2,ldots, v_n}\$ and edge set \$E(G) = {e_1, e_2,ldots, e_m}\$. Similar to the Randi'c matrix, here we introduce the Randi'c incidence matrix of a graph \$G\$, denoted by \$I_R(G)\$, which is defined as the \$ntimes m\$ matrix whose \$(i,j)\$-entry is \$(d_i)^{-frac{1}{2}}\$ if \$v_i\$ is incident to \$e_j\$ and \$0\$ otherwise. Naturally, the Randi'c incidence energy \$I_RE\$ of \$G\$ is the sum of the singular values of \$I_R(G)\$. We establish lower and upper bounds for the Randic incidence energy. Graphs for which these bounds are best possible are characterized. Moreover, we investigate the relation between the Randic incidence energy of a graph and that of its subgraphs. Also we give a sharp upper bound for the Randic incidence energy of a bipartite graph and determine the trees with the maximum Randic incidence energy among all \$n\$-vertex trees. As a result, some results are very different from those for incidence energy. https://toc.ui.ac.ir/article_5573_68f2261c2087d1f09fb34c2f8de4b053.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 3 4 2014 12 01 On Lict sigraphs 11 18 5627 10.22108/toc.2014.5627 EN Veena Mathad University of Mysore Kishori P. Narayankar Mangalore University Journal Article 2013 06 20 A signed graph (marked graph) is an ordered pair \$S=(G,sigma)\$‎ ‎\$(S=(G,mu))\$‎, ‎where \$G=(V,E)\$ is a graph called the underlying‎ ‎graph of \$S\$ and \$sigma:Erightarrow{+,-}\$‎ ‎\$(mu:Vrightarrow{+,-})\$ is a function‎. ‎For a graph \$G\$‎, ‎\$V(G)‎, ‎E(G)\$ and \$C(G)\$ denote its vertex set‎, ‎edge set and cut-vertex‎ ‎set‎, ‎respectively‎. ‎The lict graph \$L_{c}(G)\$ of a graph \$G=(V,E)\$‎ ‎is defined as the graph having vertex set  \$E(G)cup C(G)\$ in which‎ ‎two vertices are adjacent if and only if they correspond to‎ ‎adjacent edges of \$G\$ or one corresponds to an edge \$e_{i}\$ of \$G\$‎ ‎and the other corresponds to a cut-vertex \$c_{j}\$ of \$G\$ such that‎ ‎\$e_{i}\$ is incident with \$c_{j}\$‎. ‎In this paper‎, ‎we introduce lict‎ ‎sigraphs‎, ‎as a natural extension of the notion of lict graph to‎ ‎the realm of signed graphs‎. ‎We show that every lict sigraph is‎ ‎balanced‎. ‎We characterize signed graphs \$S\$ and \$S^{'}\$ for which‎ ‎\$Ssim L_{c}(S)\$‎, ‎\$eta(S)sim L_{c}(S)\$‎, ‎\$L(S)sim L_{c}(S')\$‎, ‎\$J(S)sim L_{c}(S^{'})\$ and \$T_{1}(S)sim L_{c}(S^{'})\$‎, ‎where‎ ‎\$eta(S)\$‎, ‎\$L(S)\$‎, ‎\$J(S)\$ and \$T_{1}(S)\$ are negation‎, ‎line graph‎, ‎jump graph and semitotal line sigraph of \$S\$‎, ‎respectively‎, ‎and‎ ‎\$sim\$ means switching equivalence‎. https://toc.ui.ac.ir/article_5627_e7de2aef7c26e21d97bfaf79f2112406.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 3 4 2014 12 01 The geodetic domination number for the product of graphs 19 30 5750 10.22108/toc.2014.5750 EN S. Robinson Chellathurai Scott Christian College S. Padma Vijaya University College of Engineering Nagercoil Journal Article 2014 03 06 A subset \$S\$ of vertices in a graph \$G\$ is called a geodetic set if every vertex not in \$S\$ lies on a shortest path between two vertices from \$S\$‎. ‎A subset \$D\$ of vertices in \$G\$ is called dominating set if every vertex not in \$D\$ has at least one neighbor in \$D\$‎. ‎A geodetic dominating set \$S\$ is both a geodetic and a dominating set‎. ‎The geodetic (domination‎, ‎geodetic domination) number \$g(G) (gamma(G),gamma_g(G))\$ of \$G\$ is the minimum cardinality among all geodetic (dominating‎, ‎geodetic dominating) sets in \$G\$‎. ‎In this paper‎, ‎we show that if a triangle free graph \$G\$ has minimum degree at least 2 and \$g(G) = 2\$‎, ‎then \$gamma _g(G) = gamma(G)\$‎. ‎It is shown‎, ‎for every nontrivial connected graph \$G\$ with \$gamma(G) = 2\$ and \$diam(G) > 3\$‎, ‎that \$gamma_g(G) > g(G)\$‎. ‎The lower bound for the geodetic domination number of Cartesian product graphs is proved‎. ‎Geodetic domination number of product of cycles (paths) are determined‎. In this work‎, ‎we also determine some bounds and exact values of the geodetic domination number of strong product of graphs‎. https://toc.ui.ac.ir/article_5750_87bdcf395fa6e3fd7e39a154bc0f1442.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 3 4 2014 12 01 Comparing the second multiplicative Zagreb coindex with some graph invariants 31 41 5951 10.22108/toc.2014.5951 EN Farzaneh Falahati Nezhad Science and Research Branch, Islamic Azad University Ali Iranmanesh Department of Mathematics, Tarbiat Modares University, P. O. Box 14115-137, Tehran Abolfazl Tehranian Science and Research Branch, Islamic Azad University Mahdieh Azari Kazerun Branch, Islamic Azad University Journal Article 2014 07 16 ‎‎The second multiplicative Zagreb coindex of a simple graph \$G\$ is‎ ‎defined as‎: ‎\$\${overline{Pi{}}}_2left(Gright)=prod_{uvnotin{}E(G)}d_Gleft(uright)d_Gleft(vright),\$\$‎ ‎where \$d_Gleft(uright)\$ denotes the degree of the vertex \$u\$ of‎ ‎\$G\$‎. ‎In this paper‎, ‎we compare \$overline{{Pi}}_2\$-index with‎ ‎some well-known graph invariants such as the Wiener index‎, ‎Schultz‎ ‎index‎, ‎eccentric connectivity index‎, ‎total eccentricity‎, ‎eccentric-distance sum‎, ‎the first Zagreb index and coindex and the‎ ‎first multiplicative Zagreb index and coindex‎. https://toc.ui.ac.ir/article_5951_503474fbf2c1d8a206d3e6dd20f8a32e.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 3 4 2014 12 01 Perfect state transfer in unitary Cayley graphs over local rings 43 54 5974 10.22108/toc.2014.5974 EN Yotsanan Meemark Chulalongkorn University Songpon Sriwongsa Chulalongkorn University Journal Article 2014 03 21 ‎In this work‎, ‎using eigenvalues and eigenvectors of unitary Cayley graphs over finite local rings and elementary linear algebra‎, ‎we characterize which local rings allowing PST occurring in its unitary Cayley graph‎. ‎Moreover‎, ‎we have some developments when \$R\$ is a product of local rings‎. https://toc.ui.ac.ir/article_5974_ee7986b6514f7db7db00ebf91f7de927.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 3 4 2014 12 01 Complete solution to a conjecture of Zhang-Liu-Zhou 55 58 5986 10.22108/toc.2014.5986 EN Mostafa Tavakoli Ferdowsi University of Mashhad F. Rahbarnia Ferdowsi University of Mashhad M. Mirzavaziri Ferdowsi University of Mashhad A. R. Ashrafi University of Kashan Journal Article 2014 01 20 ‎‎Let \$d_{n,m}=big[frac{2n+1-sqrt{17+8(m-n)}}{2}big]\$ and‎ ‎\$E_{n,m}\$ be the graph obtained from a path‎ ‎\$P_{d_{n,m}+1}=v_0v_1 cdots v_{d_{n,m}}\$ by joining each vertex of‎ ‎\$K_{n-d_{n,m}-1}\$ to \$v_{d_{n,m}}\$ and \$v_{d_{n,m}-1}\$‎, ‎and by‎ ‎joining \$m-n+1-{n-d_{n,m}choose 2}\$ vertices of \$K_{n-d_{n,m}-1}\$‎ ‎to \$v_{d_{n,m}-2}\$‎. ‎Zhang‎, ‎Liu and Zhou [On the maximal eccentric‎ ‎connectivity indices of graphs‎, ‎Appl‎. ‎Math‎. ‎J‎. ‎Chinese Univ.‎, ‎in‎ ‎press] conjectured that if \$d_{n,m}geqslant 3\$‎, ‎then \$E_{n,m}\$‎ ‎is the graph with maximal eccentric connectivity index among all‎ ‎connected graph with \$n\$ vertices and \$m\$ edges‎. ‎In this note‎, ‎we‎ ‎prove this conjecture‎. ‎Moreover‎, ‎we present the graph with‎ ‎maximal eccentric connectivity index among the connected graphs‎ ‎with \$n\$ vertices‎. ‎Finally‎, ‎the minimum of this graph invariant‎ ‎in the classes of tricyclic and tetracyclic graphs are computed‎. https://toc.ui.ac.ir/article_5986_740f215bc6659e95ccaa77c44e50e504.pdf