A refinement of the unit and unitary Cayley graphs of a finite ring
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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.
Abstract
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.
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