Spectral Analysis of Compressed Zero Divisor Graphs Over \(\Pi^n_{k_=1}\) \(\mathbb{Z}_{pk}\) for 2 ≤ n ≤ 5, Where Each pk is a Prime

This paper investigates the spectral characteristics and energy parameters of compressed zero-divisor graphs corresponding to product rings of the form \(\Pi^n_{k_=1}\) \(\mathbb{Z}_{pk}\) for 2 ≤ n ≤ 5, Where Each pk is a Prime. The eigenvalue spectrum,determinant,trace,spectral radius,and energy indices of the Adjacency, Laplacian, and Seidel matrices are computed and compared. For \(\mathbb{Z}_{p1}\) x \(\mathbb{Z}_{p2}\), all three Adjacency, Laplacian, and Seidel energies were equal to 2, reflecting spectral symmetry in the simplest case. As the product expanded, a notable escalation in the energies was observed. The spectral radius ρ showed a parallel growth, indicating increasing graph complexity with higher product. Across all product rings, the Laplacian and Seidel energies consistently exceeded the adjacency energy, showing that larger ring