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.