The First General Zagreb Index of the Zero Divisor Graph for the Ring Zpqk

This study investigates the application of graph theory in analyzing the zero divisor graph of a commutative ring, with a specific focus on its connection to the topological index. For an undirected graph Γ with consists of a non-empty set of vertices, V , and a set of edges, E, the first general Zagreb index is defined as a graph invariant that measures the sum of the degree of each vertex to the power of α= 0. Meanwhile, the zero divisor graph Γ of the commutative ring, R is the (undirected) graph with vertices the zero-divisors of R, and distinct vertices a and b are adjacent if and only if ab = 0. In this paper, the general formulas of the first general Zagreb index of the zero divisor graph for the ring of integers modulo pqk are computed for the cases δ = 1, 2, and 3. This research focuses on the ring defined as the integers modulo pqk, where k is a positive integer, p and q are primes p < q. Two examples are given to demonstrate the main f indings.