Non-torsion 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 T(M) be the set of all torsion elements of M and NT(M) = M − T(M) be the set of all non-torsion elements of M. The non-torsion element graph of M over R is an undirected simple graph GNT (M) with NT(M) as vertex set and any two distinct vertices x and y are adjacent if and only if x + y ∈ T(M). In this paper, we study the basic properties of the graph GNT (M). We also study the diameter and girth of GNT (M). Further, we determine the domination number and the bondage number of GNT (M). We establish a relation between diameter and domination number of GNT (M). We also establish a relation between girth and bondage number of GNT (M). We also establish a relation between girth and bondage number of GNT (M).