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2018 Finite commutative rings whose unitary Cayley graphs have positive genus
Huadong Su, Yiqiang Zhou
J. Commut. Algebra 10(2): 275-293 (2018). DOI: 10.1216/JCA-2018-10-2-275

Abstract

The unitary Cayley graph of a ring $R$ is the simple graph whose vertices are the elements of $R$, and where two distinct vertices $x$ and $y$ are linked by an edge if and only if $x-y$ is a unit in $R$. The genus of a simple graph $G$ is the smallest nonnegative integer $g$ such that $G$ can be embedded into an orientable surface $\mathbb {S}_{g}$. It is proven that, for a given positive integer $g$, there are at most finitely many finite commutative rings whose unitary Cayley graphs have genus $g$. We determine all finite commutative rings whose unitary Cayley graphs have genus 1, 2 and 3, respectively.

Citation

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Huadong Su. Yiqiang Zhou. "Finite commutative rings whose unitary Cayley graphs have positive genus." J. Commut. Algebra 10 (2) 275 - 293, 2018. https://doi.org/10.1216/JCA-2018-10-2-275

Information

Published: 2018
First available in Project Euclid: 13 August 2018

zbMATH: 06917497
MathSciNet: MR3842338
Digital Object Identifier: 10.1216/JCA-2018-10-2-275

Subjects:
Primary: 05C25 , 13A99
Secondary: 05C75 , 13M05

Keywords: complete bipartite graph , complete graph , finite commutative ring , genus , Unitary Cayley graph

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.10 • No. 2 • 2018
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