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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