On Arithmetical Structures on Complete Graphs

TL;DR

This paper characterizes the possible largest values in arithmetical structures on complete graphs, establishing bounds and prime factor conditions, and explores which primes can occur as maximum values.

Contribution

It provides bounds and prime factor conditions for the largest values in arithmetical structures on complete graphs, advancing understanding of their possible configurations.

Findings

Existence of arithmetical structures with largest value less than a certain bound

Non-existence of structures when prime factors exceed a threshold

Characterization of primes that can be largest values

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 $(n+1)^2/4$, 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 $K_n$.