Generalized Cayley graphs associated to commutative rings

Let R be a commutative ring with identity element. For a natural number n, we associate a simple graph, denoted by , with Rn⧹{0} as the vertex set and two distinct vertices X and Y in Rn being adjacent if and only if there exists an n×n lower triangular matrix A over R whose entries on the main diagonal are non-zero and such that AXT=YTor AYT=XT, where, for a matrix B, BT is the matrix transpose of B. When we consider the ring R as a semigroup with respect to multiplication, then is the usual undirected Cayley graph (over a semigroup). Hence is a generalization of Cayley graph. In this paper we study some basic properties of . We also determine all isomorphic classes of finite commutative rings whose generalized Cayley graph has genus at most three.