non-nilpotent graph of commutative rings

Let r be a commutative ring with unity. let nil(r) be the set of all nilpotent elements of r and nil(r) = r n il(r) be the set of all non-nilpotent elements of r. the non-nilpotent graph of r is a simple undirected graph gnn (r) with nil(r) as vertex set and any two distinct vertices x and y are adjacent if and only if x + y ∈ n il(r). in this paper, we introduce and discuss the basic properties of the graph gnn (r). we also study the diameter and girth of gnn (r). further, we determine the domination number and the bondage number of gnn (r). we establish a relation between diameter and domination number of gnn (r). we also establish a relation between girth and bondage number of gnn (r).