On the Zero-divisor Cayley Graph of a Finite Commutative Ring

Let R be a finite commutative ring. Let Z(R) and J(R) be the set of all zero-divisor elements and the Jacobson radical of R, respectively. The zero-divisor Cayley graph of R, denoted by ZCAY(R), is the graph obtained by setting all the elements of Z(R) to be the vertices and defining distinct vertices x and y to be adjacent if and only if x−y ∈ Z(R). The induced subgraph of ZCAY(R) on the vertex set Z(R) \ J(R) is denoted by ZCAY∗ (R). In this paper, the basic properties of ZCAY (R) and ZCAY∗ (R) are investigated and some characterization results regarding connectedness, girth and planarity of ZCAY(R) and ZCAY∗(R) are given. Finally, we study the clique number of ZCAY(R).