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

## Authors

• Ghazali Semil @ Ismail ᵃDepartment of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia; ᵇMathematical Sciences Studies, College of Computing, Informatics and Mathematics, Universiti Teknologi MARA Johor Branch, Pasir Gudang Campus, 81750, Masai, Johor, Malaysia
• Nor Haniza Sarmin Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
• Nur Idayu Alimon Mathematical Sciences Studies, College of Computing, Informatics and Mathematics, Universiti Teknologi MARA Johor Branch, Pasir Gudang Campus, 81750, Masai, Johor, Malaysia
• Fariz Maulana Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, 40132, Bandung, Indonesia

## Keywords:

Topological index, first Zagreb index, zero divisor graph, graph theory, ring theory

## 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.

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