No abstract

Abstract. Let R be a finite commutative ring with nonzero identity. We define Γ(R) 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 Γ(R) are obtained and the vertex connectivity and the edge connectivity of Γ(R) are given. Finally, by a constructive way, we determine when the graph Γ(R) is Hamiltonian. As a consequence, we show that Γ(R) has a perfect matching if and only if |R| is an even number.