On arithmetical structures on complete graphs

Abstract

An arithmetical structure on the complete graph $K_n$ with $n$ vertices is given by a collection of $n$ positive integers with no common factor, each of which divides their sum. We show that, for all positive integers $c$ less than a certain bound depending on $n$, there is an arithmetical structure on $K_n$ with largest value $c$. We also show that, if each prime factor of $c$ is greater than ${\left(n+1\right)}^2∕4$, there is no arithmetical structure on $K_n$ with largest value $c$. We apply these results to study which prime numbers can occur as the largest value of an arithmetical structure on $K_n$.