Induced subgraphs of zero-divisor graphs

TL;DR

This paper explores the properties of zero-divisor graphs of finite commutative rings, demonstrating their universality in representing all finite graphs and analyzing their structure in various classes of rings.

Contribution

It proves that zero-divisor graphs of certain rings are universal for finite graphs and characterizes when these graphs are threshold graphs, including embeddings of complex graphs like the Rado graph.

Findings

Zero-divisor graphs of finite rings are universal for all finite graphs.

Zero-divisor graphs of local rings with principal maximal ideals are threshold graphs.

There exists a local ring whose zero-divisor graph embeds the Rado graph.

Abstract

The zero-divisor graph of a finite commutative ring with unity is the graph whose vertex set is the set of zero-divisors in the ring, with $a$ and $b$ adjacent if $ab=0$. We show that the class of zero-divisor graphs is universal, in the sense that every finite graph is isomorphic to an induced subgraph of a zero-divisor graph. This remains true for various restricted classes of rings, including boolean rings, products of fields, and local rings. But in more restricted classes, the zero-divisor graphs do not form a universal family. For example, the zero-divisor graph of a local ring whose maximal ideal is principal is a threshold graph; and every threshold graph is embeddable in the zero-divisor graph of such a ring. More generally, we give necessary and sufficient conditions on a non-local ring for which its zero-divisor graph to be a threshold graph. In addition, we show that there is a countable local ring whose zero-divisor graph embeds the Rado graph, and hence every finite or countable graph, as induced subgraph. Finally, we consider embeddings in related graphs such as the $2$-dimensional dot product graph.