Let G = (V,E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V − S, there exists u ∈ S such that d(u, v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the ...

In this article, we continue our very recent work by extending it to the complex case. Having been inspired real Hopfield neural network (HNN) results, investigations here yield various novel some of which are as follows. First, “biased pseudo-cut” concept HNN (CHNN) case, introduce a “shadow-cut” that is defined sum intercluster phased edges. Second, while discrete-time strictly minimizes in e...

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 ...

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. In this paper we introduce a new graph associated to modules over commutative rings. We study the relationship between the algebraic properties of modules and their associated graphs. A topological characterization for the completeness of the special subgraphs is presented. Also modules whose associated graph is complete...

we consider a dynamic domination problem for graphs in which an infinitesequence of attacks occur at vertices with guards and the guard at theattacked vertex is required to vacate the vertex by moving to a neighboringvertex with no guard. other guards are allowed to move at the same time, andbefore and after each attack and the resulting guard movements, the verticescontaining guards form a dom...

Metric indexing is used to organize large databases of Attributed Graphs and define fast queries on these sets. We present a proposal where indexing is based on an m-tree in which routing nodes of the m-tree act as prototypes of the sub-cluster of Attributed Graphs that routing nodes represent. In the classical schemes, the information kept in the tree nodes of the subcluster is a selected Attr...

For even q, a group G isomorphic to PSL(2, q) stabilizes a Baer conic inside a symplectic subquadrangle W(3, q) of H(3, q2). In this paper the action of G on points and lines of H(3, q2) is investigated. A construction is given of an infinite family of hyperovals of size 2(q3−q) of H(3, q2), with each hyperoval having the property that its automorphism group contains G. Finally it is shown that...

A set system is called t-intersecting if every two members meet each other in at least t elements. Katona determined the minimum ratio of the shadow and the size of such families and showed that the Erdős– Ko–Rado theorem immediately follows from this result. The aim of this note is to reproduce the proof to obtain a slight improvement in the Kneser graph. We also give a brief overview of corre...