Generalized Cayley Graph Associated to a Lattice

Let L be a lattice. For a natural number n , we associate a simple graph, denoted by Gamma(n)(L), with L-n \ {0} as the vertex set and two distinct vertices X and Y being adjacent if and only if there exists an n x n lower triangular matrix A over L such that AX(T) = Y-T or AY(T) = X-T, where for a matrix B, B-T is the matrix transpose of B In this paper, we study some basic properties of Gamma(n)(L). Also, we investigate the genus number of the generalized Cayley graph Gamma(n)(L).