The first Zagreb index of the zero divisor graph for the ring of integers modulo

The topological index provides information about the overall structure of a molecular graph and is often used in quantitative studies of chemical compounds. Consider Γ be a simple graph, consisting of a collection of edges and vertices. The first Zagreb index of a graph is calculated by adding the squared degrees of every vertex in the graph. Meanwhile, the zero divisor graph of a ring R, denoted by Γ(R), is defined as a graph where its vertices are zero divisors of R and two distinct vertices a and b are adjacent if their product is equal to zero. For q is an odd prime number and k is a positive integer, the first Zagreb index of the zero divisor graph for the commutative ring of integers modulo 2kq is determined in this paper. An example is given to illustrate the main results.