Topological Indices of Divisor Prime Graphs

TL;DR

This paper investigates the topological properties of divisor prime graphs derived from number theory, calculating various graph indices to understand their structure.

Contribution

It introduces the analysis of multiple topological indices for divisor prime graphs, a novel intersection of graph theory and number theory.

Findings

Computed Wiener, Harary, hyper-Wiener, Zagreb, Schultz, Gutman, and Eccentric connectivity indices for divisor prime graphs.

Provided insights into the structural properties of divisor prime graphs through these indices.

Abstract

Graph theory provides powerful tools for modeling concepts in number theory, leading to the introduction of graphs derived from arithmetic properties. One such structure is the divisor prime graph, $G_{Dp(n)}$. For any positive integer $n$, let $D(n)$ be the set of its positive divisors. The vertex set of $G_{Dp(n)}$ consists of the elements of $D(n)$, with the adjacency condition that two vertices $x$ and $y$ share an edge if and only if their greatest common divisor is $1$. The primary focus of this study is to evaluate the topological characteristics of $G_{Dp(n)}$. To achieve this, we analyze and compute various distance and degree-based indices, specifically focusing on the Wiener, Harary, hyper-Wiener, First and Second Zagreb, Schultz, Gutman, and Eccentric connectivity indices.