Analyzing the normalized Laplacian and Randić spectrum of cozero-divisor graph of the ring $\mathbb{Z}_n$

Abstract

In this article, we investigate the normalized Laplacian and Randić spectrum of the cozero-divisor graph of a finite commutative ring $\mathfrak{R}$ with identity $1\neq 0$. Let $Z'(\mathfrak{R})$ be the set of non-unit and non-zero elements of ring $\mathfrak{R}$. The cozero-divisor graph of $\mathfrak{R}$, denoted by $\Gamma'(\mathfrak{R})$, is a simple undirected graph having vertex set $Z'(\mathfrak{R})$ and two distinct vertices $u$ and $v$ are joined by an edge if and only if $u\notin v\mathfrak{R}$ and $v\notin u\mathfrak{R}$, where $\alpha \mathfrak{R}$ is the ideal generated by the element $\alpha$ in $\mathfrak{R}$. Specifically, we describe the normalized Laplacian spectrum and Randić spectrum of the graph $\Gamma'(\mathbb{Z}_n)$ for various values of $n$.