A REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF A FINITE RING

Title & Authors
A REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF A FINITE RING
Naghipour, Ali Reza; Rezagholibeigi, Meysam;

Abstract
Let R be a finite commutative ring with nonzero identity. We define $\small{{\Gamma}(R)}$ 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 $\small{{\Gamma}(R)}$ are obtained and the vertex connectivity and the edge connectivity of $\small{{\Gamma}(R)}$ are given. Finally, by a constructive way, we determine when the graph $\small{{\Gamma}(R)}$ is Hamiltonian. As a consequence, we show that $\small{{\Gamma}(R)}$ has a perfect matching if and only if $\small{{\mid}R{\mid}}$ is an even number.
Keywords
Cayley graphs;Clique number;Hamiltonian graphs;finite rings;matchings;unit graphs;
Language
English
Cited by
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