Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 2538-2136 3 2 2018 12 01 Classification of rings with toroidal annihilating-ideal graph 93 119 EN Selvakumar Krishnan Department of Mathematics Manonmaniam Sundaranar University Tirunelveli selva_158@yahoo.co.in Subbulakshmi P Manonmaniam Sundaranar University shunlaxmi@gmail.com 10.22049/cco.2018.26060.1072 Let R be a non-domain commutative ring with identity and A(R) be the<br />set of non-zero ideals with non-zero annihilators. We call an ideal I of R, an<br />annihilating-ideal if there exists a non-zero ideal J of R such that IJ = (0).<br />The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex<br />set A(R) and two distinct vertices I and J are adjacent if and only if IJ =<br />(0). In this paper, we characterize all commutative Artinian nonlocal rings R<br />for which AG(R) has genus one. annihilating-ideal,planar,genus,local ring,annihilating-ideal graph http://comb-opt.azaruniv.ac.ir/article_13745.html http://comb-opt.azaruniv.ac.ir/article_13745_89fafddf8b6d794a4500e5751f76a3bc.pdf
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 2538-2136 3 2 2018 12 01 On the harmonic index of bicyclic graphs 121 142 EN Reza Rasi Azarbaijan Shahid Madani University r.rasi@azaruniv.edu 10.22049/cco.2018.26171.1081 The harmonic index of a graph \$G\$, denoted by \$H(G)\$, is defined as<br />the sum of weights \$2/[d(u)+d(v)]\$ over all edges \$uv\$ of \$G\$, where<br />\$d(u)\$ denotes the degree of a vertex \$u\$. Hu and Zhou [Y. Hu and X. Zhou, WSEAS Trans. Math. {bf 12} (2013) 716--726] proved that for any bicyclic graph \$G\$ of order \$ngeq 4\$, \$H(G)le frac{n}{2}-frac{1}{15}\$ and characterize all extremal bicyclic graphs.<br />In this paper, we prove that for any bicyclic graph \$G\$ of order \$ngeq 4\$ and maximum degree \$Delta\$, \$\$frac{1}{2} H(G)le left{begin{array}{ll}<br />frac{3n-1}{6} & {rm if}; Delta=4<br />&<br />frac{2Delta-n-3}{Delta+1}+frac{n-Delta+3}{Delta+2}+frac{1}{2}+frac{n-Delta-1}{3} & {rm if};Deltage 5 ;{rm and}; nle 2Delta-4<br />&<br />frac{Delta}{Delta+2}+frac{Delta-4}{3}+frac{n-2Delta+4}{4} & {rm if};Deltage 5 ;{rm and};nge 2Delta-3,<br />end{array}right.\$\$<br />and characterize all extreme bicyclic graphs. harmonic index,bicyclic graphs,trees http://comb-opt.azaruniv.ac.ir/article_13746.html http://comb-opt.azaruniv.ac.ir/article_13746_f0c613a9e6610951d57150aad863731f.pdf
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 2538-2136 3 2 2018 12 01 Complexity and approximation ratio of semitotal domination in graphs 143 150 EN Zehui Shao Guangzhou University zshao@gzhu.edu.cn Pu Wu Guangzhou University wupu@mail.cdu.edu.cn 10.22049/cco.2018.25987.1065 A set \$S subseteq V(G)\$ is a semitotal dominating set of a graph \$G\$ if<br /> it is a dominating set of \$G\$ and<br />every vertex in \$S\$ is within distance 2 of another vertex of \$S\$. The<br />semitotal domination number \$gamma_{t2}(G)\$ is the minimum<br />cardinality of a semitotal dominating set of \$G\$.<br />We show that the semitotal domination problem is<br />APX-complete for bounded-degree graphs, and the semitotal domination problem in any graph of maximum degree \$Delta\$ can be approximated with an approximation<br />ratio of \$2+ln(Delta-1)\$. semitotal domination,APX-complete,NP-completeness http://comb-opt.azaruniv.ac.ir/article_13748.html http://comb-opt.azaruniv.ac.ir/article_13748_70d5d03f125812cbc3dc8d0aec38312f.pdf
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 2538-2136 3 2 2018 12 01 Some results on a supergraph of the comaximal ideal graph of a commutative ring 151 172 EN S. Visweswaran Saurashtra University s_visweswaran2006@yahoo.co.in Jaydeep Parejiya Department of Mathematics, Saurashtra University, Rajkot, Gujarat, India. parejiyajay@gmail.com 10.22049/cco.2018.26132.1079 Let R be a commutative ring with identity such that R admits at least two maximal ideals. In this article, we associate a graph with R whose vertex set is the set of all proper ideals I of R such that I is not contained in the Jacobson radical of R and distinct vertices I and J are joined by an edge if and only if I and J are not comparable under the inclusion relation. The aim of this article is to study the interplay between the graph-theoretic properties of this graph and the ring-theoretic properties of the ring R. Chained ring,Bipartite graph,Split graph,Complemented graph http://comb-opt.azaruniv.ac.ir/article_13778.html http://comb-opt.azaruniv.ac.ir/article_13778_c5b20d65e49415f10224ec5da091faf6.pdf
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 2538-2136 3 2 2018 12 01 Lower bounds on the signed (total) \$k\$-domination number 173 178 EN Lutz Volkmann 0000-0003-3496-277X RWTH Aachen University volkm@math2.rwth-aachen.de 10.22049/cco.2018.26055.1071 Let \$G\$ be a graph with vertex set \$V(G)\$. For any integer \$kge 1\$, a signed (total) \$k\$-dominating function<br />is a function \$f: V(G) rightarrow { -1, 1}\$ satisfying \$sum_{xin N[v]}f(x)ge k\$ (\$sum_{xin N(v)}f(x)ge k\$)<br />for every \$vin V(G)\$, where \$N(v)\$ is the neighborhood of \$v\$ and \$N[v]=N(v)cup{v}\$. The minimum of the values<br />\$sum_{vin V(G)}f(v)\$, taken over all signed (total) \$k\$-dominating functions \$f\$, is called the signed (total)<br />\$k\$-domination number. The clique number of a graph \$G\$ is the maximum cardinality of a complete subgraph of \$G\$.<br />In this note we present some new sharp lower bounds on the signed (total) \$k\$-domination number<br />depending on the clique number of the graph. Our results improve some known bounds. signed \$k\$-dominating function,signed \$k\$-domination number,clique number http://comb-opt.azaruniv.ac.ir/article_13779.html http://comb-opt.azaruniv.ac.ir/article_13779_039e0161b2a16abce42b7a252a65cb4e.pdf
Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 2538-2136 3 2 2018 12 01 Leap Zagreb indices of trees and unicyclic graphs 179 194 EN Ivan Gutman University of Kragujevac gutman@kg.ac.rs Zehui Shao Guangzhou University zshao@gzhu.edu.cn Zepeng Li Lanzhou University lizp@lzu.edu.cn ShaohuiShaohui Wang Department of Mathematics and Computer Science, Adelphi University, Garden City, NY, USA. shaohuiwang@yahoo.com Pu We Guangzhou University, puwu1997@126.com 10.22049/cco.2018.26285.1092 By d(v|G) and d_2(v|G) are denoted the number of first and second neighbors<br />of the vertex v of the graph G. The first, second, and third leap Zagreb indices<br />of G are defined as<br />LM_1(G) = sum_{v in V(G)} d_2(v|G)^2, LM_2(G) = sum_{uv in E(G)} d_2(u|G) d_2(v|G),<br />and LM_3(G) = sum_{v in V(G)} d(v|G) d_2(v|G), respectively. In this paper, we generalize<br />the results of Naji et al. [Commun. Combin. Optim. 2 (2017), 99-117], pertaining to <br />trees and unicyclic graphs. In addition, we determine upper and lower bounds<br />for these leap Zagreb indices and characterize the extremal graphs. Leap Zagreb index,Zagreb index,degree (of vertex) http://comb-opt.azaruniv.ac.ir/article_13782.html http://comb-opt.azaruniv.ac.ir/article_13782_6ae3457e7f09b8f6c913dd0fa53fa742.pdf