Unified A_α-spectral analysis and Wiener bounds of the essential ideal graph over ℤ_n

Abstract

This paper investigates the spectral and structural properties of the essential ideal graph E_Z_n over the ring Z_n within the unified A_α-matrix framework. Explicit expressions for the A_α-spectrum of E_Z_n are derived for several classes of integers n, including prime powers n=p^m, biprime cases n=pq, and mixed factorizations such as n=p^mq. Eigenvalues, multiplicities, and spectral bounds are established using decomposition techniques and equitable partitions. In addition, the Wiener index of E_Z_n is studied, with exact computations for small values of n and new bounds obtained in terms of spectral parameters and degree sequences. These results demonstrate the interplay between the ring-theoretic structure of Z_n and the spectral characteristics of its associated essential ideal graph.