Zagreb indices of the generalized zero-divisor graph of the ring $\mathbb{Z}_n$

Source abstract

In this paper, we compute the first and second Zagreb indices and the first and second multiplicative Zagreb indices of the generalized zero-divisor graph $\Gamma'(\mathbb{Z}_n)$, where $n = pq$, $p^\alpha q$ and $p^\alpha q^\beta$, with $p$ and $q$ distinct primes and $\alpha, \beta$ positive integers. For each case, we construct an explicit partition of the vertex set, determine the degree of each vertex class, and derive closed-form expressions for all four Zagreb indices. These indices are important in chemical graph theory and network analysis, providing quantitative measures of the structural characteristics of the graph. Received: 19 June 2026 | Revised: 12 August 2026 | Accepted: 14 August 2026