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

Abstract

Let Γ be a simple graph with the set of vertices and edges. The first Zagreb index of a graph is defined as the sum of the degree of each vertex to the power of two. Meanwhile, the zero divisor graph of a ring R, denoted by Γ(R), is defined as a graph with its vertex set Z(R)*contains the nonzero zero divisors in which two distinct vertices u and v are adjacent if uv=vu=0. In this paper, the general formula of the first Zagreb index of the zero divisor graph for the commutative ring of integers modulo pk, Z pk where a prime number p and a positive integer k is determined. A A few examples are given to illustrate the main results.