Zagreb Indices and Co-indices in 2-idempotent Divisor Graph

Abstract

This paper studies the Zagreb indices and co-indices of the graph  

   , where is a direct product of two local ring of nilpotency 2. It focuses on cases where one component is involution, and on the general case where neither component is. Explicit formulas are derived based on the number of units in each component, and examples are given to support and illustrate the result