Classification of rings with toroidal annihilating-ideal graph
Selvakumar
Krishnan
Department of Mathematics
Manonmaniam Sundaranar University
Tirunelveli
author
Subbulakshmi
P
Manonmaniam Sundaranar University
author
text
article
2018
eng
Let R be a non-domain commutative ring with identity and A(R) be theset of non-zero ideals with non-zero annihilators. We call an ideal I of R, anannihilating-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 vertexset 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 Rfor which AG(R) has genus one.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
3
v.
2
no.
2018
93
119
http://comb-opt.azaruniv.ac.ir/article_13745_89fafddf8b6d794a4500e5751f76a3bc.pdf
dx.doi.org/10.22049/cco.2018.26060.1072
On the harmonic index of bicyclic graphs
Reza
Rasi
Azarbaijan Shahid Madani University
author
text
article
2018
eng
The harmonic index of a graph $G$, denoted by $H(G)$, is defined asthe 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 $Delta$, $$frac{1}{2} H(G)le left{begin{array}{ll}frac{3n-1}{6} & {rm if}; Delta=4\&\frac{2Delta-n-3}{Delta+1}+frac{n-Delta+3}{Delta+2}+frac{1}{2}+frac{n-Delta-1}{3} & {rm if};Deltage 5 ;{rm and}; nle 2Delta-4\&\frac{Delta}{Delta+2}+frac{Delta-4}{3}+frac{n-2Delta+4}{4} & {rm if};Deltage 5 ;{rm and};nge 2Delta-3,\end{array}right.$$and characterize all extreme bicyclic graphs.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
3
v.
2
no.
2018
121
142
http://comb-opt.azaruniv.ac.ir/article_13746_f0c613a9e6610951d57150aad863731f.pdf
dx.doi.org/10.22049/cco.2018.26171.1081
Complexity and approximation ratio of semitotal domination in graphs
Zehui
Shao
Guangzhou University
author
Pu
Wu
Guangzhou University
author
text
article
2018
eng
A set $S subseteq V(G)$ is a semitotal dominating set of a graph $G$ if it is a dominating set of $G$ andevery vertex in $S$ is within distance 2 of another vertex of $S$. Thesemitotal domination number $gamma_{t2}(G)$ is the minimumcardinality of a semitotal dominating set of $G$.We show that the semitotal domination problem isAPX-complete for bounded-degree graphs, and the semitotal domination problem in any graph of maximum degree $Delta$ can be approximated with an approximationratio of $2+ln(Delta-1)$.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
3
v.
2
no.
2018
143
150
http://comb-opt.azaruniv.ac.ir/article_13748_70d5d03f125812cbc3dc8d0aec38312f.pdf
dx.doi.org/10.22049/cco.2018.25987.1065
Some results on a supergraph of the comaximal ideal graph of a commutative ring
S.
Visweswaran
Saurashtra University
author
Jaydeep
Parejiya
Department of Mathematics, Saurashtra University, Rajkot, Gujarat, India.
author
text
article
2018
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
3
v.
2
no.
2018
151
172
http://comb-opt.azaruniv.ac.ir/article_13778_c5b20d65e49415f10224ec5da091faf6.pdf
dx.doi.org/10.22049/cco.2018.26132.1079
Lower bounds on the signed (total) $k$-domination number
Lutz
Volkmann
RWTH Aachen University
author
text
article
2018
eng
Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis 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 numberdepending on the clique number of the graph. Our results improve some known bounds.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
3
v.
2
no.
2018
173
178
http://comb-opt.azaruniv.ac.ir/article_13779_039e0161b2a16abce42b7a252a65cb4e.pdf
dx.doi.org/10.22049/cco.2018.26055.1071
Leap Zagreb indices of trees and unicyclic graphs
Ivan
Gutman
University of Kragujevac
author
Zehui
Shao
Guangzhou University
author
Zepeng
Li
Lanzhou University
author
ShaohuiShaohui
Wang
Department of Mathematics and Computer Science, Adelphi University,
Garden City, NY, USA.
author
Pu
We
Guangzhou University,
author
text
article
2018
eng
By d(v|G) and d_2(v|G) are denoted the number of first and second neighborsof the vertex v of the graph G. The first, second, and third leap Zagreb indicesof G are defined asLM_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 generalizethe results of Naji et al. [Commun. Combin. Optim. 2 (2017), 99-117], pertaining to trees and unicyclic graphs. In addition, we determine upper and lower boundsfor these leap Zagreb indices and characterize the extremal graphs.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
3
v.
2
no.
2018
179
194
http://comb-opt.azaruniv.ac.ir/article_13782_6ae3457e7f09b8f6c913dd0fa53fa742.pdf
dx.doi.org/10.22049/cco.2018.26285.1092