Spectral Analysis of Order GCD Graph on Integers Modulo Ring

Abstract

Let \(n\) be a positive integer and let \(\mathbb{Z}_n\) denote the ring of integers modulo \(\mathbb{Z}_n\). We introduce the order GCD graph \(\Theta_{\mathbb{Z}_n}\), whose vertex set is \(\mathbb{Z}_n\), where two distinct vertices \(a\) and \(b\) are adjacent if and only if \(gcd(|a|,|b|)=|a \cdot b|\), with \(|a|\) denoting the multiplicative order of \(a\) in \(\mathbb{Z}_n\). We investigate fundamental structural properties of \(\Theta_{\mathbb{Z}_n}\), including a spectral analysis of the graph by studying the eigenvalues of its adjacency matrix and their relationship to the arithmetic structure of \(\mathbb{Z}_n\). Several illustrative examples are provided to highlight the interplay between number-theoretic properties of \(n\) and the spectral characteristics of \(\Theta_{\mathbb{Z}_n}\).