Generalized cayley graph of upper triangular matrix rings

Abstract

Let R be a commutative ring with the non-zero identity and n be a natural number. GRn is a simple graph with Rn\{0} as the vertex set and two distinct vertices X and Y in Rn are 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 such that AXt = Yt or AYt = Xt, where, for a matrix B, Bt is the matrix transpose of B. GRn is a generalization of Cayley graph. Let Tn(R) denote the n × n upper triangular matrix ring over R. In this paper, for an arbitrary ring R, we investigate the properties of the graph GTn(R)n.