Some graph parameters of the Zero-divisor graphs of finite commutative rings

Abstract

In this paper, some graph parameters of the zero-divisor graph $\Gamma(R)$ of a finite commutative ring $R$ for $R\simeq \mathbb{Z}_p \times \mathbb{Z}_{p^2}$ and $R\simeq \mathbb{Z}_p \times \mathbb{Z}_{2p}$ where $p>2$ a prime, are investigated. The graph $\Gamma(R)$ is a simple graph whose vertex set is the set of non-zero zero-divisors of a commutative ring $R$ with non-zero identity and two vertices $u$ and $v$ are adjacent if and only if $uv=vu=0$.

In this paper, we study some of the topological indices such as graph energies, the Zagreb indices and the domination parameters of graphs $\Gamma\big(\mathbb{Z}_p \times \mathbb{Z}_{p^2} \big)$ and $\Gamma\big(\mathbb{Z}_p \times \mathbb{Z}_{2p}\big)$.