Let R be a commutative ring with identity element. For a natural number n, we associate a simple graph, denoted by Gamma^n_R , with R^n setminus { as the vertex set and two distinct vertices X and Y in R^n being adjacent if and only if there exists an n times n lower triangular matrix A over R whose entries on the main diagonal are non-zero and such that AX^T=Y^Tor AY^T=X^T, where, for a matrix B, B^T is the matrix transpose of B. When we consider the ring R as a semigroup with respect to multiplication, then Gamma^1_R is the usual undirected Cayley graph (over a semigroup). Hence Gamma^n_R is a generalization of Cayley graph. In this paper we study some basic properties of Gamma^n_R. We also determine all isomorphic classes of finite commutative rings whose generalized Cayley graph has genus at most three.

Keywords