Spectral analysis and energy bounds for total zero-divisor graphs of the ring $\mathbb{Z}_{p^2q^2}$

Abstract

This study investigates the spectral properties of the total zero-divisor graph associated with the ring $\mathbb{Z}_{p^2q^2}$, where $p$ and $q$ are distinct primes. Using an explicit equitable partition of the graph, we obtain a complete description of the adjacency spectrum. We show that the spectrum consists of two integer eigenvalues $0$ and $-1$ occurring with large multiplicities determined in closed form, together with seven eigenvalues from the quotient matrix, of which five are nonzero. The decomposition enables explicit computation and bounding of key spectral invariants. In particular, we derive new formulas and bounds for the graph energy and provide structural constraints for the spectral radius. These results deepen the understanding of total zero-divisor graphs over composite rings and contribute new tools for further developments in algebraic graph theory. Received: 29 December 2025 | Accepted: 08 May 2026