In this paper, we examine the interplay between the structural and spectral properties of the unit graph $G\left({\mathbb{Z}}_n\right)$ for $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$, where $p_1,p_2,\dots ,p_k$ are distinct primes and $k,r_1,r_2,\dots ,r_k$ are positive integers such that at least one of the $r_i$ must be greater than 1. We first analyze the structure of the unit graph of ${\mathbb{Z}}_n$, treating it as what we will define as a ‘generalized join graph’ under these conditions. We then determine the Laplacian spectrum of $G\left({\mathbb{Z}}_n\right)$ and deduce that it is integral for all n. Consequently, we obtain the Laplacian spectral radius and algebraic connectivity of $G\left({\mathbb{Z}}_n\right)$. We also prove that the vertex connectivity of $G\left({\mathbb{Z}}_{pq}\right)$ is $(p-2)q$, where $2\ne p<q$. We deduce the vertex connectivity of $G\left({\mathbb{Z}}_n\right)$ when $n=p^r{q}^s$, where $2\ne p<q$ are primes and $r,s$ are positive integers. Finally, we present conjectures regarding the vertex connectivity of $G\left({\mathbb{Z}}_n\right)$ when $n=p_1{p}_2\dots p_k$ and $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$, where $p_i$ are distinct primes, $r_i$ are positive integers, and $1\le i\le k$.

In this paper, we examine the interplay between the structural and spectral properties of the unit graph G(Zn) for n=p1r1p2r2...pkrk, where p1,p2,...,pk are distinct primes and k,r1,r2,...,rk are positive integers such that at least one of the ri must be greater than 1. We first analyze the structure of the unit graph of Zn as a generalized join graph under these conditions. We then determine the Laplacian spectrum of G(Zn) and deduce that it is integral for all n. Consequently, we obtain Laplacian spectral radius and algebraic connectivity of G(Zn). We also prove that the vertex connectivity of G(Zpq) is (p−2)q, where 2≠p<q. We deduce the vertex connectivity of G(Zn) when n=prqs, where 2≠p<q are primes and r,s are positive integers. Finally, we present conjectures about the vertex connectivity of G(Zn) when n=p1p2...pk and n=p1r1p2r2...pkrk, where pi are distinct primes, ri are positive integers, and 1≤i≤k.

For α ∈ [0, 1], the matrix defines a Laplacian–type operator associated with a graph G , interpolating between − A ( G ) and D ( G ). In this paper, we develop a systematic study of the ‐spectrum under several classical graph products. Complete spectral characterizations are obtained for the Cartesian, lexicographic, and generalized lexicographic products. For the direct and strong products of connected regular graphs, explicit formulas are derived in terms of the Laplacian and signless Laplacian spectra of the factor graphs. The results demonstrate that exhibits a spectral behavior that is structurally distinct from the classical A _α ‐matrix, particularly in the presence of product constructions. These findings provide a unified framework for the analysis of across graph products and contribute to the broader development of parametrized spectral graph theory.

This paper explores the common neighborhood energy ($E_{CN}\left(\mathsf{\Gamma }\right)$) of graphs derived from the dihedral group $D_{2n}$ and generalized quaternion group $Q_{4n}$, specifically the non-commuting graph (NCM-graph) and the commuting graph (CM-graph). Studying graphs associated with groups offers a powerful approach to translating algebraic properties into combinatorial structures, enabling the application of graph-theoretic tools to understand group behavior. The common neighborhood energy, defined as the sum of the absolute values of the eigenvalues of the common neighborhood (CN) matrix, i.e., ${\sum }_{i=1}^p\left|{\zeta }_i\right|$, where ${\left\{{\zeta }_i\right\}}_{i=1}^p$ are the CN eigenvalues, provides insights into the structural properties of these graphs. We derive explicit formulas for the CN characteristic polynomials and corresponding CN eigenvalues for both the NCM-graph and CM-graph as functions of n. Consequently, we establish closed-form expressions for the $E_{CN}$ of these graphs, which are parameterized by n. The validity of our theoretical results is confirmed through computational examples. This study contributes to the spectral analysis of algebraic graphs, demonstrating a direct connection between the group-theoretic structure of $D_{2n}$ and $Q_{4n}$, as well as the combinatorial energy of their associated graphs, thus furthering the understanding of group properties through spectral graph theory.