Integer divisor connectivity graph

Abstract

UDC 512.5

Let $n$ be a nonprime integer. We introduce a new simple undirected graph and denote it by $MD(n),$ where the vertices are the proper divisors of $n$ and two vertices $x$ and $y$ are adjacent if $xy$ divides $n.$ We explore the connectedness of $MD(n)$ and provide detailed calculations for the degree of each vertex. In addition, we focus on the special case where $n = p^{\alpha},$ where $p$ is a prime positive integer and $\alpha\geq 3$ is a positive integer. For these instances, we explicitly determine the chromatic number $\chi$ and the clique number $\omega$ of $MD(n).$ Finally, we conclude that $\chi(MD(n)) = \omega(MD(n)).$