Energy and Topological Indices of Complete Bipartite Subgraphs

This paper investigates the complete bipartite subgraphs induced within the zero-divisor graph of a commutative ring formed by the direct product of three distinct modular integer rings. The set of nonzero zero-divisors is partitioned into six disjoint subsets based on the position of the zero component in each element. Six complete bipartite subgraphs are constructed and analysed by pairing subsets with zeros in different positions. For each subgraph, we compute the energy, Laplacian energy, and three degree-based multiplicative topological indices, namely the Narumi–Katayama index, and the first and second multiplicative Zagreb indices. The results are expressed in closed-form formulas and reveal consistent structural patterns, highlighting the relationship between the algebraic properties of the ring and the graph theoretic characteristics of the induced subgraphs.