On arithmetical structures on complete graphs

Harris, Zachary; Louwsma, Joel

doi:10.2140/involve.2020.13.345

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Form

Policies for Authors

Ethics Statement

ISSN: 1944-4184 (e-only)

ISSN: 1944-4176 (print)

Author Index

Coming Soon

Other MSP Journals

This article is available for purchase or by subscription. See below.

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 $K_{n}$ .

PDF Access Denied

We have not been able to recognize your IP address 18.216.190.167 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 30.00:

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