Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
3
2
2018
12
01
Classification of rings with toroidal annihilating-ideal graph
93
119
EN
Selvakumar
Krishnan
Department of Mathematics
Manonmaniam Sundaranar University
Tirunelveli
selva_158@yahoo.co.in
Subbulakshmi
P
Manonmaniam Sundaranar University
shunlaxmi@gmail.com
10.22049/cco.2018.26060.1072
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
http://comb-opt.azaruniv.ac.ir/article_13745.html
http://comb-opt.azaruniv.ac.ir/article_13745_89fafddf8b6d794a4500e5751f76a3bc.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
3
2
2018
12
01
On the harmonic index of bicyclic graphs
121
142
EN
Reza
Rasi
Azarbaijan Shahid Madani University
r.rasi@azaruniv.edu
10.22049/cco.2018.26171.1081
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
http://comb-opt.azaruniv.ac.ir/article_13746.html
http://comb-opt.azaruniv.ac.ir/article_13746_f0c613a9e6610951d57150aad863731f.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
3
2
2018
12
01
Complexity and approximation ratio of semitotal domination in graphs
143
150
EN
Zehui
Shao
Guangzhou University
zshao@gzhu.edu.cn
Pu
Wu
Guangzhou University
wupu@mail.cdu.edu.cn
10.22049/cco.2018.25987.1065
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
http://comb-opt.azaruniv.ac.ir/article_13748.html
http://comb-opt.azaruniv.ac.ir/article_13748_70d5d03f125812cbc3dc8d0aec38312f.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
3
2
2018
12
01
Some results on a supergraph of the comaximal ideal graph of a commutative ring
151
172
EN
S.
Visweswaran
Saurashtra University
s_visweswaran2006@yahoo.co.in
Jaydeep
Parejiya
Department of Mathematics, Saurashtra University, Rajkot, Gujarat, India.
parejiyajay@gmail.com
10.22049/cco.2018.26132.1079
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
http://comb-opt.azaruniv.ac.ir/article_13778.html
http://comb-opt.azaruniv.ac.ir/article_13778_c5b20d65e49415f10224ec5da091faf6.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
3
2
2018
12
01
Lower bounds on the signed (total) $k$-domination number
173
178
EN
Lutz
Volkmann
RWTH Aachen University
volkm@math2.rwth-aachen.de
10.22049/cco.2018.26055.1071
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
http://comb-opt.azaruniv.ac.ir/article_13779.html
http://comb-opt.azaruniv.ac.ir/article_13779_039e0161b2a16abce42b7a252a65cb4e.pdf
Azarbaijan Shahid Madani University
Communications in Combinatorics and Optimization
2538-2128
2538-2136
3
2
2018
12
01
Leap Zagreb indices of trees and unicyclic graphs
179
194
EN
Ivan
Gutman
University of Kragujevac
gutman@kg.ac.rs
Zehui
Shao
Guangzhou University
zshao@gzhu.edu.cn
Zepeng
Li
Lanzhou University
lizp@lzu.edu.cn
ShaohuiShaohui
Wang
Department of Mathematics and Computer Science, Adelphi University,
Garden City, NY, USA.
shaohuiwang@yahoo.com
Pu
We
Guangzhou University,
puwu1997@126.com
10.22049/cco.2018.26285.1092
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)
http://comb-opt.azaruniv.ac.ir/article_13782.html
http://comb-opt.azaruniv.ac.ir/article_13782_6ae3457e7f09b8f6c913dd0fa53fa742.pdf