Genus Bounds for Harmonic Group Actions on Finite Graphs

TL;DR

This paper establishes sharp bounds for the size of harmonic automorphism groups on finite graphs of genus g, drawing analogies with classical results on Riemann surfaces and proving these bounds are attained infinitely often.

Contribution

It introduces the concept of harmonic group actions on graphs and proves sharp bounds for their maximal size, extending classical genus bounds to a graph-theoretic setting.

Findings

Bounds for M(g): 4(g-1) and 6(g-1)

Both bounds are sharp and attained infinitely often

Only these two values occur for M(g)

Abstract

This paper develops graph analogues of the genus bounds for the maximal size of an automorphism group of a compact Riemann surface of genus $g\ge 2$. Inspired by the work of M. Baker and S. Norine on harmonic morphisms between finite graphs, we motivate and define the notion of a harmonic group action. Denoting by M(g) the maximal size of such a harmonic group action on a graph of genus $g\ge 2$, we prove that $4(g-1)\le M(g)\le 6(g-1)$, and these bounds are sharp in the sense that both are attained for infinitely many values of g. Moreover, we show that the values $4(g-1)$ and $6(g-1)$ are the only values taken by the function $M(g)$.