Patterns of primes and composites on divisibility 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.