Cayley graphs of ideals in a commutative ring

Let R be a commutative ring. We associate a digraph to the ideals of R whose vertex set is the set of all nontrivial ideals of R and, for every two distinct vertices I and J, there is an arc from I to J, denoted by I ? J, whenever there exists a nontrivial ideal L such that J = IL. We call this graph the ideal digraph of R and denote it by ??I?(R). Also, for a semigroup H and a subset S of H, the Cayley graph Cay(H,S) of H relative to S is defined as the digraph with vertex set H and edge set E(H,S) consisting of those ordered pairs (x,y) such that y = sx for some s ? S. In fact the ideal digraph ??I?(R) is isomorphic to the Cayley graph Cay(I ? ,I ? ), where I is the set of all ideals of R and I ? consists of nontrivial ideals. The undirected ideal (simple) graph of R, denoted by I?(R), has an edge joining I and J whenever either J = IL or I = JL, for some nontrivial ideal L of R. In this paper, we study some basic properties of graphs ??I?(R) and I?(R) such as connectivity, diameter, graph height, Wiener index and clique number. Moreover, we study the Hasse ideal digraph ??H?(R), which is a spanning subgraph of ??I?(R) such that for each two distinct vertices I and J, there is an arc from I to J in ??H?(R) whenever I ? J in ??I?(R), and there is no vertex L such that I ? L and L ? J in ??I?(R).