A graph related to Euler ϕ function

2020/12/23 by Ghanbari, Nima, Alikhani, Saeid
#05C05 #11AXX #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2012.12492

Abstract

Euler function ϕ(n) is the number of positive integers less than n and relatively prime to n. Suppose that ϕ¹(n)=ϕ(n) and ϕⁱ(n)=ϕ(ϕ^i-1(n)). Let A⊆ ℕ, and A_ϕ=\ ϕ^k(n)| n∈ A , k∈ ℕ ∪ \0\\. We consider a graph G_ϕ(A)=(V,E), where V=A_ϕ and E=\\r,s\| r,s∈ V, ϕ(r)=s \. We say a graph H is a G_ϕ-graph, if there exists a set of natural numbers A, such that H=G_ϕ(A). In this paper we study the graph G_ϕ(A) and investigate some specific graphs and some chemical trees as G_ϕ-graph.

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