University of Isfahan Transactions on Combinatorics 2251-8657 4 1 2015 03 01 Restrained roman domination in graphs 1 17 4395 10.22108/toc.2015.4395 EN Roushini Leely Pushpam Department of Mathematics D.B.Jain College, Chennai 97 India Sampath Padmapriea Department of Mathematics Sri Sairam Engineering College Chennai 44 India Journal Article 2013 09 17 ‎A textit{Roman dominating function} (RDF) on a graph \$G = (V,E)\$ is‎ ‎defined to be a function \$ f:V rightarrow lbrace 0,1,2rbrace\$‎ ‎satisfying the condition that every vertex \$u\$ for which \$f(u) = 0\$ is‎ ‎adjacent to at least one vertex \$v\$ for which \$f(v)=2\$‎. ‎A set \$S‎ ‎subseteq V\$ is a textit{Restrained dominating set} if every vertex‎ ‎not in \$S\$ is adjacent to a vertex in \$S\$ and to a vertex in \$V‎ - ‎S\$‎. ‎We define a textit{Restrained Roman dominating function} on a‎ ‎graph \$G = (V,E)\$ to be a function \$f‎ : ‎V rightarrow lbrace 0,1,2‎ ‎rbrace\$ satisfying the condition that every vertex \$u\$ for which‎ ‎\$f(u) = 0 \$ is adjacent to at least one vertex \$v\$ for which \$f(v)=2\$‎ ‎and at least one vertex \$w\$ for which \$f(w) = 0\$‎. ‎The textit{weight}‎ ‎of a Restrained Roman dominating function is the value \$f(V)= sum _{u‎ ‎in V} f(u)\$‎. ‎The minimum weight of a Restrained Roman dominating‎ ‎function on a graph \$G\$ is called the <em>Restrained Roman‎ ‎domination number</em> of \$G\$ and denoted by \$gamma_{rR}(G)\$‎. ‎In this‎ ‎paper‎, ‎we initiate a study of this parameter‎. https://toc.ui.ac.ir/article_4395_4464d5648abc97e1c5cf12df1d49d67c.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 4 1 2015 03 01 Optimal orientations of subgraphs of complete bipartite graphs 19 29 4725 10.22108/toc.2015.4725 EN R. Lakshmi Assistant Professor Department of mathematics Annamalai University Annamalainagar 608002 G. Rajasekaran Research Scholar, Department of Mathematics, Annamalai University, Annamalainagar 608 002 R. Sampathkumar Mathematics Section, FEAT, Annamalai University Annamalainagar 6008002 Journal Article 2013 09 13 For a graph \$G,\$ let \$mathscr{D}(G)\$ be the set of all strong digraphs \$D\$ obtained by the orientations of \$G.\$ The <em>orientation number</em> of \$G\$ is \$stackrel{rightarrow}{d}!!(G),=\$ min \${d(D),|,D,in,mathscr{D}(G)},\$ where \$d(D)\$ denotes the diameter of the digraph \$D.\$ In this paper‎, ‎we determine the orientation number for some subgraphs of complete bipartite graphs‎. https://toc.ui.ac.ir/article_4725_ab6061f1b408a4d9a9b58cff4bb6771c.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 4 1 2015 03 01 A bound for the locating chromatic number of trees 31 41 6024 10.22108/toc.2015.6024 EN Ali Behtoei Imam Khomeini International University Mahdi Anbarloei Imam Khomeini International University Journal Article 2014 03 14 Let \$f\$ be a proper \$k\$-coloring of a connected graph \$G\$ and \$Pi=(V_1,V_2,ldots,V_k)\$ be an ordered partition of \$V(G)\$ into the resulting color classes. For a vertex \$v\$ of \$G\$, the color code of \$v\$ with respect to \$Pi\$ is defined to be the ordered \$k\$-tuple \$c_{{}_Pi}(v)=(d(v,V_1),d(v,V_2),ldots,d(v,V_k)),\$ where \$d(v,V_i)=min{d(v,x):~xin V_i}, 1leq ileq k\$. If distinct vertices have distinct color codes, then \$f\$ is called a locating coloring. The minimum number of colors needed in a locating coloring of \$G\$ is the locating chromatic number of \$G\$, denoted by \$chi_{L}(G)\$. In this paper, we study the locating chromatic  numbers of trees. We provide a counter example to a theorem of Gary Chartrand et al. [G. Chartrand, D. Erwin, M.A. Henning, P.J. Slater, P. Zhang, The locating-chromatic number of a graph, <em>Bull. Inst. Combin. Appl.,</em> <strong>36</strong> (2002) 89-101] about the locating chromatic number of trees. Also, we offer a new bound for the locating chromatic number of trees. Then, by constructing a special family of trees, we show that this bound is best possible. https://toc.ui.ac.ir/article_6024_1de06593073b82da914fe9c02657202d.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 4 1 2015 03 01 Graphs with fixed number of pendent vertices and minimal first Zagreb index 43 48 6029 10.22108/toc.2015.6029 EN Ivan Gutman University of Kragujevac Kragujevac, Serbia Muhammad Kamran Jamil Government College University Naveed Akhter Government College University Journal Article 2014 08 01 ‎The first Zagreb index \$M_1\$ of a graph \$G\$ is equal to the sum of squares‎ ‎of degrees of the vertices of \$G\$‎. ‎Goubko proved that for trees with \$n_1\$‎ ‎pendent vertices‎, ‎\$M_1 geq 9,n_1-16\$‎. ‎We show how this result can be‎ ‎extended to hold for any connected graph with cyclomatic number \$gamma geq 0\$‎. ‎In addition‎, ‎graphs with \$n\$ vertices‎, ‎\$n_1\$ pendent vertices‎, ‎cyclomatic‎ ‎number \$gamma\$‎, ‎and minimal \$M_1\$ are characterized‎. ‎Explicit expressions‎ ‎for minimal \$M_1\$ are given for \$gamma=0,1,2\$‎, ‎which directly can be extended‎ ‎for \$gamma>2\$‎. https://toc.ui.ac.ir/article_6029_3088c8b600fd6edb7a3e85ff1dd901d2.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 4 1 2015 03 01 \$k\$-odd mean labeling of prism 49 56 5388 10.22108/toc.2015.5388 EN B. Gayathri Periyar E.V.R. College(Autonomous) K. Amuthavalli Roever Engineering College Journal Article 2013 09 16 ‎A \$(p,q)\$ graph \$G\$ is said to have a \$k\$-odd mean‎ ‎labeling \$(k ge 1)\$ if there exists an injection \$f‎ : ‎V‎ ‎to {0‎, ‎1‎, ‎2‎, ‎ldots‎, ‎2k‎ + ‎2q‎ - ‎3}\$ such that the‎ ‎induced map \$f^*\$ defined on \$E\$ by \$f^*(uv) =‎ ‎leftlceil frac{f(u)+f(v)}{2}rightrceil\$ is a‎ ‎bijection from \$E\$ to \${2k‎ - ‎1‎, ‎2k‎ + ‎1‎, ‎2k‎ + ‎3‎, ‎ldots‎, ‎2‎ ‎k‎ + ‎2q‎ - ‎3}\$‎. ‎A  graph that admits \$k\$-odd mean‎ ‎labeling is called \$k\$-odd mean graph‎. ‎In this paper‎, ‎we investigate \$k\$-odd mean labeling of prism \$C_m times P_n\$‎. https://toc.ui.ac.ir/article_5388_c3d1298da0efeb3c0c28f5a89756531d.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 4 1 2015 03 01 Note on the skew energy of oriented graphs 57 61 6117 10.22108/toc.2015.6117 EN Jun He School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, PR China Ting-Zhu Huang School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P.R. China Journal Article 2014 08 04 The skew energy of oriented graphs is defined as the‎ ‎sum of the norms of all the eigenvalues of the skew  adjacency matrix‎. ‎In this note‎, ‎we obtain some upper bounds for the skew energy of any oriented graphs‎, ‎which improve the known upper‎ ‎bound obtained by Adiga et al‎. https://toc.ui.ac.ir/article_6117_67b546232db1fead989a7b5928ccfacc.pdf