Bulletin of the
Korean Mathematical Society BKMS

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

Keywords: Cayley graphs, Clique number, Hamiltonian graphs, finite rings, matchings, unit graphs

MSC numbers: 05C25, 05C40, 05C75, 13M05, 16U60