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Abstract
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An arithmetical structure on the complete graph
with
vertices is given by a
collection of
positive
integers with no common factor, each of which divides their sum. We show that, for all positive integers
less than a certain bound
depending on
, there is an
arithmetical structure on
with largest value
.
We also show that, if each prime factor of
is greater than
, there is no arithmetical
structure on
with
largest value
. We
apply these results to study which prime numbers can occur as the largest value of an arithmetical
structure on
.
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Keywords
arithmetical structure, complete graph, Diophantine
equation, Laplacian matrix, prime number
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Mathematical Subject Classification 2010
Primary: 11D68
Secondary: 05C50, 11A41
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Milestones
Received: 15 September 2019
Revised: 1 January 2020
Accepted: 6 January 2020
Published: 30 March 2020
Communicated by Joshua Cooper
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