On divisibility relation graphs

TL;DR

This paper studies the properties of divisibility relation graphs, focusing on their structural invariants, planarity, and spectral characteristics, providing new insights into their graph-theoretic features.

Contribution

It determines key invariants and properties of divisibility relation graphs, a specific class of graphs associated with divisibility partial orders.

Findings

Calculated clique and independence numbers for divisibility graphs.

Identified conditions for planarity of these graphs.

Explored spectral properties through numerical experiments.

Abstract

For each positive integer $n$, we define the divisibility relation graph $D_n$ whose vertex set is the set of divisors of $n$, and in which two vertices are adjacent if one is a divisor of the other. This type of graph is a special case of graphs associated with a partial order, which have been widely studied in the literature. In this work, we determine various graph-theoretic invariants of divisibility relation graphs, such as their clique and independence numbers, and their planarity. We also discuss various spectral properties that are discovered by our numerical experiments.