A graph related to Euler $\varphi$ function

Abstract

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

arXiv comment10 pages, 10 figures