Let $ R $ be a ring and $ U(R) $ be the set of unit elements of $ R $. The unit graph $ G(R) $ of $ R $ is the graph whose vertices are all the elements of $ R $, defining distinct vertices $ x $ and $ y $ to be adjacent if and only if $ x + y \in U(R) $. The Laplacian spectrum of $ G(\mathbb{Z}_n) $ was studied when $ n = p^{m} $, where $ p $ is a prime and $ m $ is a positive integer. Consequently, in this paper, we study the Laplacian spectrum of $ G(\mathbb{Z}_n) $, for $ n = p_1p_2...p_k $, where $ p_i $ are distinct primes and $ i = 1, 2, ..., k $.

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 $1\le i\le k$.

Let $\mathcal{Z}$ be a graph of order n with m edges. Let $A_{ag}\left(\mathcal{Z}\right)$ represents the arithmetic–geometric matrix of $\mathcal{Z}$. The eigenvalues of the matrix $A_{ag}\left(\mathcal{Z}\right)$ are called the arithmetic–geometric eigenvalues, and the eigenvalue with the largest modulus is called the arithmetic–geometric spectral radius of $\mathcal{Z}$. The sum of the absolute values of the arithmetic–geometric eigenvalues is called the arithmetic–geometric energy of $\mathcal{Z}$. In this paper, we establish sharp upper and lower bounds for the AM-GM spectral radius in terms of various graph parameters and provide a complete characterization of the extremal graphs that attain these bounds. Additionally, we derive new bounds for the AM-GM energy of a graph and identify the corresponding extremal structures. In both contexts, our results significantly improve upon several existing bounds reported in the literature.

Let $\Gamma$ be a graph of order n with m edges, and let $A_{ag}(\Gamma )$ denote its arithmetic–geometric matrix. The eigenvalues of $A_{ag}(\Gamma )$ are referred to as the arithmetic–geometric eigenvalues of $\Gamma$. The sum of the absolute values of these eigenvalues is called the arithmetic–geometric energy of $\Gamma$. In this paper, we establish some new upper and lower bounds for the arithmetic–geometric energy in terms of various topological indices, and we completely characterize the extremal graphs attaining these bounds. We show that our results significantly improve upon several existing bounds reported in the literature. We also provide several ways to construct graphs with the same AM–GM energy but different AM–GM spectra.

Let $ G $ be a graph of order $ n $ with $ m $ edges. Let $ \tilde{L}(G) $ be the Laplacian atom-bond connectivity (ABC)-matrix of $ G $. The eigenvalues of $ \tilde{L}(G) $ are the Laplacian ABC-eigenvalues of $ G $, and its largest eigenvalue is called the Laplacian ABC-spectral radius of $ G $. In this paper, we present some new upper and lower bounds for the Laplacian ABC-spectral radius, and we characterize the extremal graphs attaining these bounds. We also present an upper bound for the second smallest Laplacian ABC-eigenvalue and characterize the extremal graphs. We determine the Laplacian ABC-eigenvalues of the join of two regular graphs and the join of a regular graph with the union of two regular graphs, in terms of the Laplacian eigenvalues of the component graphs.

Let $A_{ex}\left(\mathcal{G}\right)$ be the extended adjacency matrix of G. The eigenvalues of $A_{ex}\left(\mathcal{G}\right)$ are called extended adjacency eigenvalues of $\mathcal{G}$. The sum of the absolute values of eigenvalues of the $A_{ex}$-matrix is called the extended adjacency energy ${\mathcal{E}}_{ex}\left(\mathcal{G}\right)$ of $\mathcal{G}$. In this paper, we obtain the $A_{ex}$-spectrum of the joined union of regular graphs in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix. Consequently, we derive the $A_{ex}$-spectrum of the join of two regular graphs, the lexicographic product of regular graphs, and the $A_{ex}$-spectrum of various families of graphs. Further, as applications of our results, we construct infinite classes of infinite families of extended adjacency equienergetic graphs. We show that the $A_{ex}$-energy of the join of two regular graphs is greater than or equal to their energy. We also determine the $A_{ex}$-eigenvalues of the power graph of finite abelian groups.