Journal of Commutative Algebra

Finite commutative rings whose unitary Cayley graphs have positive genus

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.

Article information

Source
J. Commut. Algebra, Volume 10, Number 2 (2018), 275-293.

Dates
First available in Project Euclid: 13 August 2018

https://projecteuclid.org/euclid.jca/1534125829

Digital Object Identifier
doi:10.1216/JCA-2018-10-2-275

Zentralblatt MATH identifier
06917497

Citation

Su, Huadong; Zhou, Yiqiang. Finite commutative rings whose unitary Cayley graphs have positive genus. J. Commut. Algebra 10 (2018), no. 2, 275--293. doi:10.1216/JCA-2018-10-2-275. https://projecteuclid.org/euclid.jca/1534125829

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