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 set of non-zero ideals with non-zero annihilators. We call an ideal $I$ of $R$, an annihilating-ideal if there exists a non-zero ideal $J$ of $R$ such that $IJ =(0)$. The annihilating-ideal graph of $R$ is defined as the graph $AG(R)$ with the vertex set $A^*(R)$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ =(0)$. In this paper, we characterize all commutative Artinian nonlocal rings $R$ for which $AG(R)$ has genus one. http://comb-opt.azaruniv.ac.ir/article_13745_64b16e21db453555d1fe39afe192d7e5.pdfAzarbaijan 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 the sum of weights $2/[d(u)+d(v)]$ over all edges $uv$ of $G$, where $d(u)$ 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. In this paper, we prove that for any bicyclic graph $G$ of order $ngeq 4$ and maximum degree $Delta$, <br />$$H(G)le left{begin{array}{ll}<br />frac{3n-1}{6} & {rm if}; Delta=4\<br />&\<br />2(frac{2Delta-n-3}{Delta+1}+frac{n-Delta+3}{Delta+2}+frac{1}{2}+frac{n-Delta-1}{3}) & {rm if};Deltage 5 ;{rm and}; nle 2Delta-4\<br />&\<br />2(frac{Delta}{Delta+2}+frac{Delta-4}{3}+frac{n-2Delta+4}{4}) & {rm if};Deltage 5 ;{rm and};nge 2Delta-3,\<br />end{array}right.$$ <br />and characterize all extreme bicyclic graphs.http://comb-opt.azaruniv.ac.ir/article_13746_f0c613a9e6610951d57150aad863731f.pdfAzarbaijan 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 it is a dominating set of $G$ and every vertex in $S$ is within distance 2 of another vertex of $S$. The semitotal domination number $gamma_{t2}(G)$ is the minimum cardinality of a semitotal dominating set of $G$. We show that the semitotal domination problem is APX-complete for bounded-degree graphs, and the semitotal domination problem in any graph of maximum degree $Delta$ can be approximated with an approximation ratio of $2+ln(Delta-1)$.http://comb-opt.azaruniv.ac.ir/article_13748_70d5d03f125812cbc3dc8d0aec38312f.pdfAzarbaijan 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 $R$.http://comb-opt.azaruniv.ac.ir/article_13778_c5b20d65e49415f10224ec5da091faf6.pdfAzarbaijan 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 is a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$) 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 $sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functions $f$, is called the signed (total) $k$-domination number. The clique number of a graph $G$ is the maximum cardinality of a complete subgraph of $G$. In this note we present some new sharp lower bounds on the signed (total) $k$-domination number depending on the clique number of the graph. Our results improve some known bounds.http://comb-opt.azaruniv.ac.ir/article_13779_039e0161b2a16abce42b7a252a65cb4e.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21283220181201Leap Zagreb indices of trees and unicyclic graphs1791941378210.22049/cco.2018.26285.1092ENIvanGutmanUniversity of Kragujevac0000-0001-9681-1550ZehuiShaoGuangzhou 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 of the vertex $v$ of the graph $G$. The first, second, and third leap Zagreb indices of $G$ are defined as $LM_1(G) = sum_{v in V(G)} d_2(v|G)^2$, $LM_2(G) = sum_{uv in E(G)} d_2(u|G),d_2(v|G)$, and $LM_3(G) = sum_{v in V(G)} d(v|G),d_2(v|G)$, respectively. In this paper, we generalize the results of Naji et al. [Commun. Combin. Optim. {bf 2} (2017), 99--117], pertaining to trees and unicyclic graphs. In addition, we determine upper and lower bounds on these leap Zagreb indices and characterize the extremal graphs.http://comb-opt.azaruniv.ac.ir/article_13782_6ae3457e7f09b8f6c913dd0fa53fa742.pdf