Laplacian spectrum of the cozero-divisor graph of ring

Barkha Baloda, Praveen Mathil, Jitender Kumar

Published 2022-02-01Version 1

Let $R$ be a ring with unity. The cozero-divisor graph of a ring $R$, denoted by $\Gamma^{'}(R)$, is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $x \notin Ry$ and $y \notin Rx$. In this paper, we study the Laplacian spectrum of $\Gamma^{'}(\mathbb{Z}_n)$. We show that the graph $\Gamma^{'}(\mathbb{Z}_{pq})$ is Laplacian integral. Further, we obtain the Laplacian spectrum $\Gamma^{'}(\mathbb{Z}_n)$ for $n = p^{n_1}q^{n_2}$, where $n_1, n_2 \in \mathbb{N}$ and $p, q$ are distinct primes.