Non-nilpotent element graph of a module over a commutative ring

Let R be a commutative ring with non-zero unity and M be a unitary R-module. Let Nil(M) be the set of all the nilpotent elements of M and Non(M)=M-Nil(M) be the set of all non-nilpotent elements of M. The non-nilpotent element graph of M over R is an undirected simple graph G_{NN}(M) with Non(M) as vertex set and any two distinct vertices x and y are adjacent if and only if x+y\in Nil(M). In this paper, we study the basic properties of the graph G_{NN}(M). We also study the diameter and girth of G_{NN}(M). Further, we determine the domination number and the bondage number of G_{NN}(M). We establish a relation between the diameter and domination number of G_{NN}(M). We also establish a relation between the girth and bondage number of G_{NN}(M).

کلیدواژه ها:

Bondage number ، diameter ، domination number ، girth ، Nilpotent elements