Generalized Cayley graphs associated to commutative rings

AbstractLet R be a commutative ring with identity element. For a natural number n, we associate a simple graph, denoted by ΓRn, 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 ΓR1 is the usual undirected Cayley graph (over a semigroup). Hence ΓRn is a generalization of Cayley graph. In this paper we study some basic properties of ΓRn. We also determine all isomorphic classes of finite commutative rings whose generalized Cayley graph has genus at most three