A Remark on the Energy Bounds and Degree-Based Topological Indices of the Zero-Divisor Graph ℤₚ[x]/⟨x⁴⟩

Abstract

The zero-divisor graph of a commutative ring, Γ(R), is a prominent subject in graph theory. Previous research has investigated the energy and topological indices of Γ(ℤₚ[x]/⟨x⁴⟩) for any prime p; however, errors were identified in the characteristic polynomial and the associated energy bounds. Recent work by Rather addressed these discrepancies using a vertex partitioning and quotient matrix approach. In contrast, this study presents a distinct matrix-theoretic approach based on the Schur complement to systematically derive the characteristic polynomial and its corresponding eigenvalues, thereby independently confirming and reinforcing the corrected energy bounds. Furthermore, this study significantly extends the existing literature by investigating a broader range of previously unexplored degree-based topological indices, including the second modified Zagreb, general and inverse general Randić, third and fifth symmetric division, harmonic, inverse sum, and forgotten indices.