A graph related to Euler $\phi$ function

TL;DR

This paper explores a graph constructed from iterated Euler phi function values, analyzing its structure and identifying specific graphs and chemical trees as instances of these $G_$-graphs.

Contribution

It introduces the concept of $G_$-graphs based on iterated Euler phi function values and studies their properties, including specific graphs and chemical trees.

Findings

Identification of certain graphs as $G_$-graphs

Analysis of the structure of $G_$-graphs

Application to chemical tree structures

Abstract

Euler function $\phi(n)$ is the number of positive integers less than $n$ and relatively prime to $n$. Suppose that $\phi^1(n)=\phi(n)$ and $\phi^i(n)=\phi(\phi^{i-1}(n))$. Let $A\subseteq \mathbb{N}$, and $A_{\phi}=\{ \phi^k(n)| n\in A , k\in \mathbb{N} \cup \{0\}\}.$ We consider a graph $G_{\phi}(A)=(V,E)$, where $V=A_{\phi}$ and $E=\{\{r,s\}| r,s\in V, \phi(r)=s \}$. We say a graph $H$ is a $G_\phi$-graph, if there exists a set of natural numbers $A$, such that $H=G_{\phi}(A)$. In this paper we study the graph $G_{\phi}(A)$ and investigate some specific graphs and some chemical trees as $G_\phi$-graph.