On the Cayley graph of a commutative ring with respect to its zero-divisors

TL;DR

This paper explores the properties of Cayley graphs constructed from the zero-divisors of a commutative ring, revealing structural insights and connectivity results, especially for finite and zero-dimensional rings.

Contribution

It introduces a detailed study of Cayley graphs of commutative rings with respect to zero-divisors, including connectivity, perfectness, and chromatic properties, extending understanding of algebraic graph structures.

Findings

For zero-dimensional non-local rings, the Cayley graph is connected with diameter 2.

Vertex and edge connectivity are determined for finite rings.

The clique and chromatic numbers of the regular element subgraph are computed for finite rings.

Abstract

Let $R$ be a commutative ring with unity and $R^{+}$ be $Z^*(R)$ be the additive group and the set of all non-zero zero-divisors of $R$, respectively. We denote by $\mathbb{CAY}(R)$ the Cayley graph $Cay(R^+,Z^*(R))$. In this paper, we study $\mathbb{CAY}(R)$. Among other results, it is shown that for every zero-dimensional non-local ring $R$, $\mathbb{CAY}(R)$ is a connected graph of diameter 2. Moreover, for a finite ring $R$, we obtain the vertex connectivity and the edge connectivity of $\mathbb{CAY}(R)$. We investigate rings $R$ with perfect $\mathbb{CAY}(R)$ as well. We also study $Reg(\mathbb{CAY}(R))$ the induced subgraph on the regular elements of $R$. This graph gives a family of vertex transitive graphs. We show that if $R$ is a Noetherian ring and $Reg(\mathbb{CAY}(R))$ has no infinite clique, then $R$ is finite. Furthermore, for every finite ring $R$, the clique number and the chromatic number of $Reg(\mathbb{CAY}(R))$ are determined.