Abstract

Consider a commutative ring $R$ with an identity element $1\neq0$. The set $Z(R)'$ is defined as the collection of all elements in the ring $R$ that are non-zero and non units. The undirected graph ${\Gamma'(R)}$ with vertex set $Z(R)'$, known as the cozero-divisor graph of $R$. In this graph, two distinct vertices, denoted by $x$ and $y$, are connected by an edge if and only if $x$ is not an element of the ideal $yR$ and $y$ is not an element of the ideal $xR$. In this article, we find the Laplacian eigenvalues of the graphs ${\Gamma'(\mathbb Z_n)}$ for $n=q_1^{Q}q_2q_3$ and $q_1^{Q_1}q_2^{Q_2}q_3$, where $q_1,q_2,q_3$ are distinct primes and $Q, {Q_1},{Q_2}$ are positive integers.

Keywords: Ring of integer modulo $n$; Laplacian matrix; Laplacian spectrum; cozero-divisor graph.

MSC: 15A18, 05C25, 05C50

DOI: 10.57016/MV-KVEL1652

Pages:  1--12