2019-02-20T06:40:06Z
http://comb-opt.azaruniv.ac.ir/?_action=export&rf=summon&issue=2265
Communications in Combinatorics and Optimization
Commun. Comb. Optim.
2538-2128
2538-2128
2018
3
2
Classification of rings with toroidal annihilating-ideal graph
Selvakumar
Krishnan
Subbulakshmi
P
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.
annihilating-ideal
planar
genus
local ring
annihilating-ideal graph
2018
12
01
93
119
http://comb-opt.azaruniv.ac.ir/article_13745_89fafddf8b6d794a4500e5751f76a3bc.pdf
Communications in Combinatorics and Optimization
Commun. Comb. Optim.
2538-2128
2538-2128
2018
3
2
On the harmonic index of bicyclic graphs
Reza
Rasi
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.
harmonic index
bicyclic graphs
trees
2018
12
01
121
142
http://comb-opt.azaruniv.ac.ir/article_13746_f0c613a9e6610951d57150aad863731f.pdf
Communications in Combinatorics and Optimization
Commun. Comb. Optim.
2538-2128
2538-2128
2018
3
2
Complexity and approximation ratio of semitotal domination in graphs
Zehui
Shao
Pu
Wu
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 $2+ln(Delta-1)$.
semitotal domination
APX-complete
NP-completeness
2018
12
01
143
150
http://comb-opt.azaruniv.ac.ir/article_13748_70d5d03f125812cbc3dc8d0aec38312f.pdf
Communications in Combinatorics and Optimization
Commun. Comb. Optim.
2538-2128
2538-2128
2018
3
2
Some results on a supergraph of the comaximal ideal graph of a commutative ring
S.
Visweswaran
Jaydeep
Parejiya
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
2018
12
01
151
172
http://comb-opt.azaruniv.ac.ir/article_13778_c5b20d65e49415f10224ec5da091faf6.pdf
Communications in Combinatorics and Optimization
Commun. Comb. Optim.
2538-2128
2538-2128
2018
3
2
Lower bounds on the signed (total) $k$-domination number
Lutz
Volkmann
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.
signed $k$-dominating function
signed $k$-domination number
clique number
2018
12
01
173
178
http://comb-opt.azaruniv.ac.ir/article_13779_039e0161b2a16abce42b7a252a65cb4e.pdf
Communications in Combinatorics and Optimization
Commun. Comb. Optim.
2538-2128
2538-2128
2018
3
2
Leap Zagreb indices of trees and unicyclic graphs
Ivan
Gutman
Zehui
Shao
Zepeng
Li
ShaohuiShaohui
Wang
Pu
We
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.
Leap Zagreb index
Zagreb index
degree (of vertex)
2018
12
01
179
194
http://comb-opt.azaruniv.ac.ir/article_13782_6ae3457e7f09b8f6c913dd0fa53fa742.pdf