On Derivative Euler Phi Function Set-Graphs

TL;DR

This paper explores the graph-theoretic properties of two new classes of graphs derived from the Euler Phi function, focusing on their structure based on divisibility and coprimality conditions.

Contribution

It introduces and analyzes derivative Euler Phi function set-graphs based on divisibility and coprimality, expanding the understanding of their structural properties.

Findings

Characterization of graph properties based on divisibility adjacency

Analysis of graph properties based on coprimality adjacency

Insights into the structural differences between the two graph types

Abstract

In this paper, we study some graph theoretical properties of two derivative Euler Phi function set-graphs. For the Euler Phi function $\phi(n)$, $n\in \mathbb{N}$, the set $S_\phi(n) =\{i:\gcd(i,n)=1, 1\leq i \leq n\}$ and the vertex set is $\{v_i:i\in S_\phi(n)\}$. Two graphs $G_d(S_\phi(n))$ and $G_p(S_\phi(n))$, defined with respect to divisibility adjacency and relatively prime adjacency conditions, are studied.