Communications in Combinatorics and Optimization
Volume:3 Issue: 2, Summer and Autumn 2018

Let R be a non-domain commutative ring with identity and A∗(R) be the set of non-zero ideals with non-zero annihilators. We call an ideal I1 of R, an annihilating-ideal if there exists a non-zero ideal I2 of R such that I1I2 = (0). The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A∗(R) and two distinct vertices I1 and I2 are adjacent if and only if I1I2 = (0). In this paper, we characterize all commutative Artinian non-local rings R for which AG(R) has genus one.

The harmonic index of a graph G, denoted by H(G), is defined as the 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. 12 (2013) 716–726] proved that for any bicyclic graph G of order n ≥ 4, H(G) ≤ n 2 − 1 15 and characterized all extremal bicyclic graphs. In this paper, we prove that for any bicyclic graph G of order n ≥ 4 and maximum degree
,
H(G) ≤ 3n−1
6 if
= 4
2( 2
−n−3
+1 + n−
+3
+2 + 1 2 + n−
−1 3 ) if
≥ 5 and n ≤ 2
− 4
2(

+2 +
−4 3 + n−2
+4 4 ) if
≥ 5 and n ≥ 2
− 3,
and characterize all extremal bicyclic graphs.

A set $S subseteq V(G)$ is a semitotal dominating set of a graph $G$ if it is a dominating set of $G$ and every vertex in $S$ is within distance 2 of another vertex of $S$. The semitotal domination number $gamma_{t2}(G)$ is the minimum cardinality of a semitotal dominating set of $G$. We show that the semitotal domination problem is APX-complete for bounded-degree graphs, and the semitotal domination problem in any graph of maximum degree $Delta$ can be approximated with an approximation ratio of $2+ln(Delta-1)$.

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.

Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating function is a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$) 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 number depending on the clique number of the graph. Our results improve some known bounds.

By d(v|G) and d_2(v|G) are denoted the number of first and second neighbors of the vertex v of the graph G. The first, second, and third leap Zagreb indices of G are defined as 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), and LM_3(G) = sum_{v in V(G)} d(v|G) d_2(v|G), respectively. In this paper, we generalize the 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 bounds for these leap Zagreb indices and characterize the extremal graphs.