Domination in the entire nilpotent element graph of a module over a commutative ring

Abstract

Let R be a commutative ring with unity and M be a unitary R module. Let Nil(M) be the set of all nilpotent elements of M. The entire nilpotent element graph of M over R is an undirected graph E(G(M)) with vertex set as M and any two distinct vertices x and y are adjacent if and only if x + y ∈ Nil(M). In this paper we attempt to study the domination in the graph E(G(M)) and investigate the domination number as well as bondage number of E(G(M)) and its induced subgraphs N(G(M)) and Non(G(M)). Some domination parameters of E(G(M)) are also studied. It has been showed that E(G(M)) is excellent, domatically full and well covered under certain conditions.

On domination in the total torsion element graph of a module

“…But not much research has been done on the domination parameters of graphs associated to algebraic structures such as groups, rings, modules in terms of algebraic properties. However, some works on domination of graphs associated to rings and modules have appeared recently, for instance see, [9,12,17,18,20,21].…”