On arithmetical structures on complete graphs

Harris, Zachary; Louwsma, Joel

doi:10.2140/involve.2020.13.345

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On arithmetical structures on complete graphs

Zachary Harris and Joel Louwsma

Vol. 13 (2020), No. 2, 345–355

DOI: 10.2140/involve.2020.13.345

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 $K_{n}$ .

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

Vol. 13, No. 2, 2020