Note on generalized Cayley graph of finite rings and its complement

Abstract

Let R be a finite commutative ring with nonzero identity and U(R) be the set of all units of R. The graph $$\Gamma =\Gamma (R,U(R),U(R))$$ 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 give a necessary and sufficient condition for $$\Gamma $$ to be unicyclic or split or claw-free. Also, we give a necessary and sufficient condition for $$\overline{\Gamma }$$ to be claw-free or unicyclic or pancyclic.