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
.
Keywords
arithmetical structure, complete graph, Diophantine
equation, Laplacian matrix, prime number