ORIGINAL_ARTICLE
Classification of rings with toroidal annihilating-ideal graph
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.
http://comb-opt.azaruniv.ac.ir/article_13745_89fafddf8b6d794a4500e5751f76a3bc.pdf
2018-12-01T11:23:20
2019-06-19T11:23:20
93
119
10.22049/cco.2018.26060.1072
annihilating-ideal
planar
genus
local ring
annihilating-ideal graph
Selvakumar
Krishnan
selva_158@yahoo.co.in
true
1
Department of Mathematics
Manonmaniam Sundaranar University
Tirunelveli
Department of Mathematics
Manonmaniam Sundaranar University
Tirunelveli
Department of Mathematics
Manonmaniam Sundaranar University
Tirunelveli
LEAD_AUTHOR
Subbulakshmi
P
shunlaxmi@gmail.com
true
2
Manonmaniam Sundaranar University
Manonmaniam Sundaranar University
Manonmaniam Sundaranar University
AUTHOR
ORIGINAL_ARTICLE
On the harmonic index of bicyclic graphs
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.
http://comb-opt.azaruniv.ac.ir/article_13746_f0c613a9e6610951d57150aad863731f.pdf
2018-12-01T11:23:20
2019-06-19T11:23:20
121
142
10.22049/cco.2018.26171.1081
harmonic index
bicyclic graphs
trees
Reza
Rasi
r.rasi@azaruniv.edu
true
1
Azarbaijan Shahid Madani University
Azarbaijan Shahid Madani University
Azarbaijan Shahid Madani University
LEAD_AUTHOR
ORIGINAL_ARTICLE
Complexity and approximation ratio of semitotal domination in graphs
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 $2+ln(Delta-1)$.
http://comb-opt.azaruniv.ac.ir/article_13748_70d5d03f125812cbc3dc8d0aec38312f.pdf
2018-12-01T11:23:20
2019-06-19T11:23:20
143
150
10.22049/cco.2018.25987.1065
semitotal domination
APX-complete
NP-completeness
Zehui
Shao
zshao@gzhu.edu.cn
true
1
Guangzhou University
Guangzhou University
Guangzhou University
LEAD_AUTHOR
Pu
Wu
wupu@mail.cdu.edu.cn
true
2
Guangzhou University
Guangzhou University
Guangzhou University
AUTHOR
ORIGINAL_ARTICLE
Some results on a supergraph of the comaximal ideal graph of a commutative ring
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.
http://comb-opt.azaruniv.ac.ir/article_13778_c5b20d65e49415f10224ec5da091faf6.pdf
2018-12-01T11:23:20
2019-06-19T11:23:20
151
172
10.22049/cco.2018.26132.1079
Chained ring
Bipartite graph
Split graph
Complemented graph
S.
Visweswaran
s_visweswaran2006@yahoo.co.in
true
1
Saurashtra University
Saurashtra University
Saurashtra University
LEAD_AUTHOR
Jaydeep
Parejiya
parejiyajay@gmail.com
true
2
Department of Mathematics, Saurashtra University, Rajkot, Gujarat, India.
Department of Mathematics, Saurashtra University, Rajkot, Gujarat, India.
Department of Mathematics, Saurashtra University, Rajkot, Gujarat, India.
AUTHOR
ORIGINAL_ARTICLE
Lower bounds on the signed (total) $k$-domination number
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.
http://comb-opt.azaruniv.ac.ir/article_13779_039e0161b2a16abce42b7a252a65cb4e.pdf
2018-12-01T11:23:20
2019-06-19T11:23:20
173
178
10.22049/cco.2018.26055.1071
signed $k$-dominating function
signed $k$-domination number
clique number
Lutz
Volkmann
volkm@math2.rwth-aachen.de
true
1
RWTH Aachen University
RWTH Aachen University
RWTH Aachen University
LEAD_AUTHOR
ORIGINAL_ARTICLE
Leap Zagreb indices of trees and unicyclic graphs
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.
http://comb-opt.azaruniv.ac.ir/article_13782_6ae3457e7f09b8f6c913dd0fa53fa742.pdf
2018-12-01T11:23:20
2019-06-19T11:23:20
179
194
10.22049/cco.2018.26285.1092
Leap Zagreb index
Zagreb index
degree (of vertex)
Ivan
Gutman
gutman@kg.ac.rs
true
1
University of Kragujevac
University of Kragujevac
University of Kragujevac
LEAD_AUTHOR
Zehui
Shao
zshao@gzhu.edu.cn
true
2
Guangzhou University
Guangzhou University
Guangzhou University
AUTHOR
Zepeng
Li
lizp@lzu.edu.cn
true
3
Lanzhou University
Lanzhou University
Lanzhou University
AUTHOR
ShaohuiShaohui
Wang
shaohuiwang@yahoo.com
true
4
Department of Mathematics and Computer Science, Adelphi University,
Garden City, NY, USA.
Department of Mathematics and Computer Science, Adelphi University,
Garden City, NY, USA.
Department of Mathematics and Computer Science, Adelphi University,
Garden City, NY, USA.
AUTHOR
Pu
We
puwu1997@126.com
true
5
Guangzhou University,
Guangzhou University,
Guangzhou University,
AUTHOR