2020-08-05T03:06:34Z
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 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
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 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
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<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
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 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
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 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)
2018
12
01
179
194
http://comb-opt.azaruniv.ac.ir/article_13782_6ae3457e7f09b8f6c913dd0fa53fa742.pdf