Communications in Combinatorics and OptimizationCommunications in Combinatorics and Optimization
http://comb-opt.azaruniv.ac.ir/
Wed, 21 Nov 2018 11:19:06 +0100FeedCreatorCommunications in Combinatorics and Optimization
http://comb-opt.azaruniv.ac.ir/
Feed provided by Communications in Combinatorics and Optimization. Click to visit.Classification of rings with toroidal annihilating-ideal graph
http://comb-opt.azaruniv.ac.ir/article_13745_2257.html
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.Fri, 30 Nov 2018 20:30:00 +0100Strong Alliances in Graphs
http://comb-opt.azaruniv.ac.ir/article_13785_0.html
For any simple connected graph $G=(V,E)$, a defensive alliance is a subset $S$ of $V$ satisfying the condition that every vertex $vin S$ has at most one more neighbour in $V-S$ than it has in $S$. The minimum cardinality of any defensive alliance in $G$ is called the alliance number of $G$, denoted $a(G)$. In this paper, we introduce a new type of alliance number called $k$-strong alliance number and its varieties. The bounds for 1-strong alliance number in terms of different graphical parameters are determined and the characterizations of graphs with 1-strong alliance number 1, 2, and $n$ are obtained.Sat, 18 Aug 2018 19:30:00 +0100On the harmonic index of bicyclic graphs
http://comb-opt.azaruniv.ac.ir/article_13746_2257.html
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.Fri, 30 Nov 2018 20:30:00 +0100New skew equienergetic oriented graphs
http://comb-opt.azaruniv.ac.ir/article_13786_0.html
Let $S(G^{sigma})$ be the skew-adjacency matrix of the oriented graph $G^{sigma}$, which is obtained from a simple undirected graph $G$ by assigning an orientation $sigma$ to each of its edges. The skew energy of an oriented graph $G^{sigma}$ is defined as the sum of absolute values of all eigenvalues of $S(G^{sigma})$. Two oriented graphs are said to be skew equienergetic iftheir skew energies are equal. In this paper, we determine the skew spectra of some new oriented graphs. As applications, we give somenew methods to construct new non-cospectral skew equienergetic oriented graphs.Thu, 16 Aug 2018 19:30:00 +0100Complexity and approximation ratio of semitotal domination in graphs
http://comb-opt.azaruniv.ac.ir/article_13748_2257.html
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 $2+ln(Delta-1)$.Fri, 30 Nov 2018 20:30:00 +0100Eternal m-security subdivision numbers in graphs
http://comb-opt.azaruniv.ac.ir/article_13803_0.html
An eternal $m$-secure set of a graph $G = (V,E)$ is aset $S_0subseteq V$ that can defend against any sequence ofsingle-vertex attacks by means of multiple-guard shifts along theedges of $G$. A suitable placement of the guards is called aneternal $m$-secure set. The eternal $m$-security number$sigma_m(G)$ is the minimum cardinality among all eternal$m$-secure sets in $G$. An edge $uvin E(G)$ is subdivided if wedelete the edge $uv$ from $G$ and add a new vertex $x$ and twoedges $ux$ and $vx$. The eternal $m$-security subdivision number${rm sd}_{sigma_m}(G)$ of a graph $G$ is the minimum cardinalityof a set of edges that must be subdivided (where each edge in $G$can be subdivided at most once) in order to increase the eternal$m$-security number of $G$. In this paper, we study the eternal$m$-security subdivision number in trees. In particular, we showthat the eternal $m$-security subdivision number of trees is atmost 2 and we characterize all trees attaining this bound.Wed, 31 Oct 2018 20:30:00 +0100Some results on a supergraph of the comaximal ideal graph of a commutative ring
http://comb-opt.azaruniv.ac.ir/article_13778_2257.html
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.Fri, 30 Nov 2018 20:30:00 +0100On the inverse maximum perfect matching problem under the bottleneck-type Hamming distance
http://comb-opt.azaruniv.ac.ir/article_13804_0.html
Given an undirected network G(V,A,c) and a perfect matching M of G, the inverse maximum perfect matching problem consists of modifying minimally the elements of c so that M becomes a maximum perfect matching with respect to the modified vector. In this article, we consider the inverse problem when the modifications are measured by the weighted bottleneck-type Hamming distance. We propose an algorithm based on the binary search technique for solving the problem. Our proposed algorithm has a better time complexity than the one presented in cite{Liu}. We also study the inverse assignment problem as a special case of the inverse maximum perfect matching problem in which the network is bipartite and present an efficient algorithm for solving the problem. Finally, we compare the algorithm with those presented in the literature.Wed, 31 Oct 2018 20:30:00 +0100Lower bounds on the signed (total) $k$-domination number
http://comb-opt.azaruniv.ac.ir/article_13779_2257.html
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.Fri, 30 Nov 2018 20:30:00 +0100Leap Zagreb indices of trees and unicyclic graphs
http://comb-opt.azaruniv.ac.ir/article_13782_2257.html
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.Fri, 30 Nov 2018 20:30:00 +0100