Topological Indices of Divisor Prime Graphs

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.