Patterns of primes and composites on divisibility graph

TL;DR

This paper analyzes the divisibility graph of consecutive natural numbers up to N, deriving formulas for graph measures and exploring patterns related to primes and composites as N varies.

Contribution

It provides analytical expressions for graph measures in the divisibility graph, highlighting how these depend on vertex labels and graph size, with special focus on prime vertices.

Findings

Formulas for degree, clustering, and centrality in divisibility graphs

Patterns in local and global measures as graph size increases

Distinct behavior of prime vertices in the graph

Abstract

We study the undirected divisibility graph in which the vertex set is a finite subset of consecutive natural numbers up to N.We derive analytical expressions for measures of the graph like degree, clustering, geodesic distance and centrality in terms of the floor functions and the divisor functions. We discuss how these measures depend on the vertex labels and the size of graph N. We also present the specific case of prime vertices separately as corollaries. We could explain the patterns in the local measures for a finite size graph as well as the trends in global measures as the size of the graph increases.