The First Zagreb Index of the Zero Divisor Graph for the Ring of Integers Modulo Power of Primes

Abstract

Let 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 , denoted by , is defined as a graph with its vertex set contains the nonzero zero divisors in which two distinct vertices and are adjacent if . In this paper, the general formula of the first Zagreb index of the zero divisor graph for the commutative ring of integers modulo , where a prime number and a positive integer is determined. A few examples are given to illustrate the main results.