Some Aspects of Zero-divisor Graph of \(\Gamma(\mathbb{Z}_{p}\times\mathbb{Z}_{p}), \Gamma(\mathbb{Z}_{p}^{m})\)
Kalyani Kalaskar *
School of Mathematical Sciences (DST-FIST), Swami Ramanand Teerth Marathwada University, Nanded–431606, Maharashtra, India.
S. M. Jogdand
Department of Mathematics, S.S.G.M. College, Loha, Swami Ramanand Teerth Marathwada University, Nanded, Maharashtra, India.
*Author to whom correspondence should be addressed.
Abstract
This paper studies the spectral properties and energy measures of zero-divisor graphs associated with the rings \(\mathbb{Z}_{p}\times\mathbb{Z}_{p}\) and \(\mathbb{Z}_{p}^{m}\) where p is a prime and m ≥ 2. using the underlying algebraic structure of these rings, we construct the corresponding adjacency and Laplacian matrices are constructed and their spectra are determined explicitly. These results lead to closed-form expressions for both adjacency energy and Laplacian energy. The analysis shows how the parameters p and m influence the spectral characteristics of the graphs, highlights the roles of symmetry and structural regularity. Furthermore, the results extend known findings for the two-dimensional case to higher-dimensional settings, thereby a broader understanding of spectral behaviour in zero-divisor graphs.
Keywords: Zero-divisor graph, adjacency matrix, Laplacian matrix, Seidel matrix, adjacency spectrum, Laplacian spectrum, Seidel Laplacian spectrum, graph energy, Laplacian energy, finite ring, spectral graph theory