Generalized unit and unitary Cayley graphs of finite rings

Let R be a finite commutative ring with nonzero identity and U(R) be the set of all units of R. The graph Gamma is the simple undirected 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 in U(R) such that x + uy is a unit in R. In this paper, we obtain degree of all vertices in Gamma and in turn provide a necessary and sufficient condition for Gamma to be Eulerian. Also, we give a necessary and sufficient condition for the complement (Gamma) over bar to be Eulerian, Hamiltonian and planar.