On arithmetical structures on K9

TL;DR

This work investigates arithmetical structures on the complete graph $K_9$ by translating the problem into finding integer solutions to the Diophantine equation $(E)_n=\sum_{i=1}^n 1/x_i=1$. It develops a $p$-adic valuation framework to constrain the solution space, and couples this with tree and automata techniques to enumerate restricted families of solutions for $n=9$ and to count solutions for general $n\ge 9$ using Mathematica and CAML programs. The paper delivers a complete restricted enumeration for $K_9$, identifies two-prime families and higher-structure patterns under $\alpha_2\le 2$, and provides recurrences and computational tools to count distinct-solution cases for larger $n$. It closes with open questions on higher $p$-adic valuations and broader two-prime divisor counts, offering a roadmap for extending these techniques to broader graph families and $n$.

Abstract

We study the arithmetical structures on the complete graph $K_9$. Our method is based on studying the solutions to writing the unit as a sum of 9 unit fractions. We work from the perspective of the Diophantine equation and use some elementary properties on the $p$-adic valuations. The proofs are assisted by trees and automata.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 1
• Lemma 1
• Corollary 1
• Corollary 2
• Corollary 3
• Corollary 4
• Corollary 5