Vol. 13, No. 2, 2020

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On arithmetical structures on complete graphs

Zachary Harris and Joel Louwsma

Vol. 13 (2020), No. 2, 345–355
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 ${K}_{n}$.

Keywords
arithmetical structure, complete graph, Diophantine equation, Laplacian matrix, prime number
Mathematical Subject Classification 2010
Primary: 11D68
Secondary: 05C50, 11A41