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
S. G. Jakkewad *
Department of Mathematics, K. B. P. College Vashi, Navi Mumbai, India.
R. G. Metkar
Department of Mathematics, Indira Gandhi (Sr) College Cidco, Nanded, India.
Y. A. Yadav
Department of Mathematics, K. B. P. College Vashi, Navi Mumbai, India.
*Author to whom correspondence should be addressed.
Abstract
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
Keywords: Compressed zero-divisor graph, product ring, adjacency matrix, Laplacian matrix, Seidel matrix, spectral properties