Malaysian Journal of Mathematical Sciences, March 2025, Vol. 19, No. 1
The Distance-Based Topological Indices of the Zero Divisor Graph for Some Commutative Rings with the Calculator App
Semil @ Ismail, G., Sarmin, N. H., and Alimon, N. I.
Corresponding Email: [email protected]
Received date: 18 April 2024
Accepted date: 7 October 2024
Abstract:
A topological index is a mathematical expression that applies to any graph that represents a molecular structure. The Harary index, Wiener index and hyper-Wiener index are distance-based topological indices of a graph. These indices use integration values to represent normalised sums of distances from a given vertex to all other vertices in the graph. The vertices represent the zero divisors of a ring, and two vertices are adjacent if their product equals zero. In this paper, these topological indices of the zero divisor graph are computed for some commutative rings $\mathbb{Z}_{p^k}$ and $\mathbb{Z}_{pq}$ where $p < q$ are primes and $k\in\mathbb{N}$ by deriving their general formulas. Several examples are provided to illustrate the main theorems. In the end, the calculator app is created by using MATLAB App Designer to compute the edges, vertices and their distance-based topological indices of the zero divisor graph for some commutative rings.
Keywords: topological index; Harary index; zero divisor graph; commutative ring; Wiener index; calculator apps; MATLAB app designer; Hyper-Wiener index
