Analysis of the zagreb indices over the zero-divisor graph of Γ(Zp × Zp),Γ(Zm p)

Abstract

We study degree based topological indices of the zero-divisor graph Γ(Zp×Zp), where p is a prime number. We first analyze the adjacency structure and degree distribution of the graph and show that Γ(Zp × Zp) is isomorphic to the complete bipartite graph Kp−1,p−1. Using this characterization, we derive explicit closed-form expressions for the first and second Zagreb indices, their corresponding coindices, as well as the multiplicative Zagreb indices and coindices. Furthermore, we introduce the Zagreb matrix associated with Γ(Zp × Zp), (Γ(Zmp) and examine its spectral properties. By determining the eigenvalues of the Zagreb matrix, we compute the Zagreb energy of the graph and express it in terms of the prime p. The results provide a complete analytical description of Zagreb-type invariants for this class of zero divisor graphs and contribute to the study of algebraically defined graphs and their degree-based invariants.