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.