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http://dx.doi.org/10.4134/BKMS.b150601

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

Naghipour, Ali Reza (Department of Mathematical Sciences Shahrekord University)
Rezagholibeigi, Meysam (Department of Mathematical Sciences Shahrekord University)
Publication Information
Bulletin of the Korean Mathematical Society / v.53, no.4, 2016 , pp. 1197-1211 More about this Journal
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 ${\mid}R{\mid}$ is an even number.
Keywords
Cayley graphs; Clique number; Hamiltonian graphs; finite rings; matchings; unit graphs;
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