Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21283220181201Classification of rings with toroidal annihilating-ideal graph931191374510.22049/cco.2018.26060.1072ENSelvakumarKrishnanDepartment of Mathematics
Manonmaniam Sundaranar University
TirunelveliSubbulakshmiPManonmaniam Sundaranar UniversityJournal Article20171020Let 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.Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21283220181201On the harmonic index of bicyclic graphs1211421374610.22049/cco.2018.26171.1081ENRezaRasiAzarbaijan Shahid Madani UniversityJournal Article20170910The 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.Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21283220181201Complexity and approximation ratio of semitotal domination in graphs1431501374810.22049/cco.2018.25987.1065ENZehuiShaoGuangzhou UniversityPuWuGuangzhou UniversityJournal Article20170729A 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 $2+ln(Delta-1)$.Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21283220181201Some results on a supergraph of the comaximal ideal graph of a commutative ring1511721377810.22049/cco.2018.26132.1079ENS.VisweswaranSaurashtra UniversityJaydeepParejiyaDepartment of Mathematics, Saurashtra University, Rajkot, Gujarat, India.Journal Article20180122Let 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.Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21283220181201Lower bounds on the signed (total) $k$-domination number1731781377910.22049/cco.2018.26055.1071ENLutzVolkmannRWTH Aachen University0000-0003-3496-277XJournal Article20171018Let $G$ 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.Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21283220181201Leap Zagreb indices of trees and unicyclic graphs1791941378210.22049/cco.2018.26285.1092ENIvanGutmanUniversity of KragujevacZehuiShaoGuangzhou UniversityZepengLiLanzhou UniversityShaohuiShaohuiWangDepartment of Mathematics and Computer Science, Adelphi University,
Garden City, NY, USA.PuWeGuangzhou University,Journal Article20180601By 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.