Genus Bounds for Harmonic Group Actions on Finite Graphs

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 $M(g)$.