# Chromatic uniqueness of zero-divisor graphs

## DOI:

https://doi.org/10.26493/2590-9770.1544.f34## Keywords:

Zero-Divisor Graph, Chromatic Equivalence, Chromatic Uniqueness## Abstract

The *zero-divisor graph* *Π*(*R*) of a commutative ring *R* is the graph whose vertices are the elements of *R* such that the vertices *u* and *v* are adjacent if and only if *u**v* = 0. If the graphs *G* and *H* have the same chromatic polynomial, then we say that they are *chromatically equivalent (or χ−equivalent)*, written as *G* ∼ *H*. Suppose a graph is uniquely determined by its chromatic polynomial. Then it is said to be *chromatically unique (or χ-unique)*.

In this paper, we discuss the question: For which numbers *n* is the graph *Π*(*Z*_{n}) *χ*-unique?

While *Z*_{n} is one of the simplest rings, we proved that for any graph *A*_{0}, for some *n*, *Π*(*Z*_{n}) contains an induced subgraph isomorphic to *A*_{0}. The first result in the subject states that for *n* ≥ 10 even, *Π*(*Z*_{n}) is not *χ*-unique (Gehet, Khalaf). By definition, *n* is *square-free* if it is prime or the product of different prime numbers. Our main result is the following. If *n* ≥ 10 is neither square-free nor the square of a prime then it is not *χ*-unique. Here and in our preceding work, we use a common method.

For odd square-free non-prime *n*, the problem is open, though on the structure of *Π*(*Z*_{n}) we know much in this case.