On $k$-dprime Divisor Function Graph

TL;DR

This paper introduces the $k$-dprime divisor function graph, extending the semiprime divisor graph, and analyzes its topological indices and properties, providing new insights into divisor function graphs.

Contribution

The paper defines the $k$-dprime divisor function graph and investigates its distance-based and degree-based topological indices, extending previous semiprime divisor graph studies.

Findings

Determined topological indices of the $k$-dprime divisor function graph.

Extended the concept of semiprime divisor graphs to $k$-dprime graphs.

Presented open problems for future research.

Abstract

Let $p$ and $q$ be distinct primes. The \textit{semiprime divisor function graph} denoted by $G_{D(pq)}$, is the graph with vertex set $V(G_{D(pq)})=\{1,p,q,pq\}$ and edge set $E(G_{D(pq)})=\{\{1,p\}, \{1,q\},\{1,pq\},\{p,pq\},\{q,pq\}\}$. The semiprime divisor function graph is a special type of divisor function graph $G_{D(n)}$ in which $n=pq$. Recently, the energy and some indices of semiprime divisor function graph have been determined. In this paper, we introduce a natural extension to the semiprime divisor function graph which we call the \textit{$k$-dprime divisor function graph}. Moreover, we present results on some distance-based and degree-based topological indices of $k$-dprime divisor function graph. We end the paper by giving some open problems.