2018
3
2
0
102
Classification of rings with toroidal annihilatingideal graph
2
2
Let R be a nondomain commutative ring with identity and A(R) be theset of nonzero ideals with nonzero annihilators. We call an ideal I of R, anannihilatingideal if there exists a nonzero ideal J of R such that IJ = (0).The annihilatingideal 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.
1

93
119


Selvakumar
Krishnan
Department of Mathematics
Manonmaniam Sundaranar University
Tirunelveli
Department of Mathematics
Manonmaniam Sundaranar
India
selva_158@yahoo.co.in


Subbulakshmi
P
Manonmaniam Sundaranar University
Manonmaniam Sundaranar University
India
shunlaxmi@gmail.com
annihilatingideal
planar
genus
local ring
annihilatingideal graph
On the harmonic index of bicyclic graphs
2
2
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) 716726] 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 $Delta$, $$frac{1}{2} H(G)le left{begin{array}{ll}frac{3n1}{6} & {rm if}; Delta=4&frac{2Deltan3}{Delta+1}+frac{nDelta+3}{Delta+2}+frac{1}{2}+frac{nDelta1}{3} & {rm if};Deltage 5 ;{rm and}; nle 2Delta4&frac{Delta}{Delta+2}+frac{Delta4}{3}+frac{n2Delta+4}{4} & {rm if};Deltage 5 ;{rm and};nge 2Delta3,end{array}right.$$and characterize all extreme bicyclic graphs.
1

121
142


Reza
Rasi
Azarbaijan Shahid Madani University
Azarbaijan Shahid Madani University
Iran
r.rasi@azaruniv.edu
harmonic index
bicyclic graphs
trees
Complexity and approximation ratio of semitotal domination in graphs
2
2
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 isAPXcomplete for boundeddegree graphs, and the semitotal domination problem in any graph of maximum degree $Delta$ can be approximated with an approximationratio of $2+ln(Delta1)$.
1

143
150


Zehui
Shao
Guangzhou University
Guangzhou University
China
zshao@gzhu.edu.cn


Pu
Wu
Guangzhou University
Guangzhou University
China
wupu@mail.cdu.edu.cn
semitotal domination
APXcomplete
NPcompleteness
Some results on a supergraph of the comaximal ideal graph of a commutative ring
2
2
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 graphtheoretic properties of this graph and the ringtheoretic properties of the ring R.
1

151
172


S.
Visweswaran
Saurashtra University
Saurashtra University
India
s_visweswaran2006@yahoo.co.in


Jaydeep
Parejiya
Department of Mathematics, Saurashtra University, Rajkot, Gujarat, India.
Department of Mathematics, Saurashtra University,
India
parejiyajay@gmail.com
Chained ring
Bipartite graph
Split graph
Complemented graph
Lower bounds on the signed (total) $k$domination number
2
2
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.
1

173
178


Lutz
Volkmann
RWTH Aachen University
RWTH Aachen University
Germany
volkm@math2.rwthaachen.de
signed $k$dominating function
signed $k$domination number
clique number
Leap Zagreb indices of trees and unicyclic graphs
2
2
By d(vG) and d_2(vG) 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(vG)^2, LM_2(G) = sum_{uv in E(G)} d_2(uG) d_2(vG),and LM_3(G) = sum_{v in V(G)} d(vG) d_2(vG), respectively. In this paper, we generalizethe results of Naji et al. [Commun. Combin. Optim. 2 (2017), 99117], pertaining to trees and unicyclic graphs. In addition, we determine upper and lower boundsfor these leap Zagreb indices and characterize the extremal graphs.
1

179
194


Ivan
Gutman
University of Kragujevac
University of Kragujevac
Serbia
gutman@kg.ac.rs


Zehui
Shao
Guangzhou University
Guangzhou University
China
zshao@gzhu.edu.cn


Zepeng
Li
Lanzhou University
Lanzhou University
Cocos (Keeling) Isles
lizp@lzu.edu.cn


ShaohuiShaohui
Wang
Department of Mathematics and Computer Science, Adelphi University,
Garden City, NY, USA.
Department of Mathematics and Computer Science,
United States
shaohuiwang@yahoo.com


Pu
We
Guangzhou University,
Guangzhou University,
China
puwu1997@126.com
Leap Zagreb index
Zagreb index
degree (of vertex)