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 Kn 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 Kn with largest value c. We also show that, if each prime factor of c is greater than (n + 1)24, there is no arithmetical structure on Kn with largest value c. We apply these results to study which prime numbers can occur as the largest value of an arithmetical structure on Kn.

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Keywords
arithmetical structure, complete graph, Diophantine equation, Laplacian matrix, prime number
Mathematical Subject Classification 2010
Primary: 11D68
Secondary: 05C50, 11A41
Milestones
Received: 15 September 2019
Revised: 1 January 2020
Accepted: 6 January 2020
Published: 30 March 2020

Communicated by Joshua Cooper
Authors
Zachary Harris
Department of Mathematics
Niagara University
Niagara University, NY
United States
Joel Louwsma
Department of Mathematics
Niagara University
Niagara University, NY
United States