Let R be a commutative ring with identity $1\ne 0$ and let $Z{\left(R\right)}^{\prime }$ be the set of all non-unit and non-zero elements of ring R. ${\Gamma }^{\prime }\left(R\right)$ denotes the cozero-divisor graph of R and is an undirected graph with vertex set $Z{\left(R\right)}^{\prime }$, $w\notin zR$, and $z\notin wR$ if and only if two distinct vertices w and z are adjacent, where $qR$ is the ideal generated by the element q in R. In this article, we investigate the signless Laplacian eigenvalues of the graphs ${\Gamma }^{\prime }\left({\mathbb{Z}}_n\right)$. We also show that the cozero-divisor graph ${\Gamma }^{\prime }\left({\mathbb{Z}}_{p_1{p}_2}\right)$ is a signless Laplacian integral.

Let us consider the finite commutative ring R, whose unity is $1\ne 0.$ Its weakly zero-divisor graph, represented as $W\Gamma \left(R\right)$, is a basic undirected graph with two distinct vertices, $c_1$ and $c_2$, that are adjacent if and only if there exist $r\in$ $\mathrm{ann}\left(c_1\right)$ and $s\in$ $\mathrm{ann}\left(c_2\right)$ that satisfy the condition $rs=0$. Let $D\left(G\right)$ be the distance matrix and $Tr\left(G\right)$ be the diagonal matrix of the vertex transmissions in basic undirected connected graph G. The $D_{\alpha }$ matrix of graph G is defined as $D_{\alpha }\left(G\right)=\alpha Tr\left(G\right)+(1-\alpha )D\left(G\right)$ for $\alpha \in [0,1]$. This article finds the $D_{\alpha }$ spectrum for the graph $W\Gamma \left({\mathbb{Z}}_n\right)$ for various values of n and also shows that $W\Gamma \left({\mathbb{Z}}_n\right)$ for $n={\vartheta }_1{\vartheta }_2{\vartheta }_3\dots {\vartheta }_t{\eta }_1^{d_1}{\eta }_2^{d_2}\dots {\eta }_s^{d_s}(d_i\ge 2,t\ge 1,s\ge 0)$, where ${\vartheta }_i$’s and ${\eta }_i$’s are the distinct primes, is $D_{\alpha }$ integral.

Let us consider the finite commutative ring R, whose unity is 1≠0. Its weakly zero-divisor graph, represented as WΓ(R), is a basic undirected graph with two distinct vertices, c1 and c2 that are adjacent if and only if there exist r∈ ann(c1) and s∈ ann(c2) that satisfy the condition rs=0. Let D(G) be the distance matrix and Tr(G) be the diagonal matrix of the vertex transmissions, in basic undirected connected graph G. The Dα matrix of graph G is defined as Dα(G)=αTr(G)+(1−α)D(G) for α∈[0,1]. This article find the Dα spectrum for the graph WΓ(Zn) for various value of n and also shows that WΓ(Zn) for n=ϑ1ϑ2ϑ3⋯ϑtη1d1η2d2⋯ηsds(di≥2,t≥1,s≥0) where ϑi’s and ηi’s are the distinct primes is Dα integral.

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.