Maximal harmonic group actions on finite graphs

TL;DR

This paper characterizes the largest possible groups acting harmonically on finite graphs, showing they are finite quotients of the modular group, and connects these findings to Hurwitz groups and surfaces.

Contribution

It provides a complete classification of maximal harmonic graph groups as quotients of the modular group, linking discrete graph symmetries to classical complex analysis.

Findings

Maximal harmonic graph groups are exactly finite quotients of the modular group.

Every Hurwitz group is a maximal graph group.

Established a connection between maximal graphs and Hurwitz surfaces.

Abstract

This paper studies groups of maximal size acting harmonically on a finite graph. Our main result states that these maximal graph groups are exactly the finite quotients of the modular group $\Gamma=\left<x,y \ | \ x^2=y^3=1\right>$ of size at least 6. This characterization may be viewed as a discrete analogue of the description of Hurwitz groups as finite quotients of the $(2,3,7)$-triangle group in the context of holomorphic group actions on Riemann surfaces. In fact, as an immediate consequence of our result, every Hurwitz group is a maximal graph group, and the final section of the paper establishes a direct connection between maximal graphs and Hurwitz surfaces via the theory of combinatorial maps.