A REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF A FINITE RING

Abstract

Let R be a finite commutative ring with nonzero identity. We define <TEX>${\Gamma}(R)$</TEX> to be the graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u of R such that x + uy is a unit of R. This graph provides a refinement of the unit and unitary Cayley graphs. In this paper, basic properties of <TEX>${\Gamma}(R)$</TEX> are obtained and the vertex connectivity and the edge connectivity of <TEX>${\Gamma}(R)$</TEX> are given. Finally, by a constructive way, we determine when the graph <TEX>${\Gamma}(R)$</TEX> is Hamiltonian. As a consequence, we show that <TEX>${\Gamma}(R)$</TEX> has a perfect matching if and only if <TEX>${\mid}R{\mid}$</TEX> is an even number.