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 .
"On arithmetical structures on complete graphs." Involve 13 (2) 345 - 355, 2020. https://doi.org/10.2140/involve.2020.13.345