On the eigenvalues of zero-divisor graph associated to finite commutative ring

Abstract

Let Z(R) be the set of zero-divisors of a commutative ring R with non-zero identity and $Z^{*}(R)=Z(R)\setminus \left\{0\right\}$ be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by $\Gamma (R),$ is a simple graph whose vertex set is $Z^{*}(R)$ and two vertices $u,v\in Z^{*}(R)$ are adjacent if and only if $uv=vu=0.$ In this paper, we investigate the adjacency matrix and the spectrum of the zero-divisor graphs $\Gamma ({\mathbb{Z}}_n)$ for $n=p^M{q}^N,$ where $p<q$ are primes and M, N are positive integers. Moreover, we obtain the clique number, stability number and girth of $\Gamma ({\mathbb{Z}}_{p^M{q}^N}).$

Keywords:

• Adjacency matrix
• zero-divisor graph
• finite commutative ring
• Euler's totient function

AMS SUBJECT CLASSIFICATION:

• 05C50
• 05C12
• 15A18

Additional information

Funding

The research of S. Pirzada is supported by the SERB-DST research project number MTR/2017/000084. Also the research of Bilal Ahmad Wani is supported by Dr. D.S. Kothari Post-Doctoral Fellowship Scheme Award Letter No. F.4-2/2006 (BSR)/MA/18-19/0037.