Genus Bounds for Harmonic Group Actions on Finite Graphs

This paper develops graph analogs of the genus bounds for the maximal size of an automorphism group of a compact Riemann surface of genus g≥2. Inspired by the work of Baker and 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≥2, we prove that 4(g−1)≤M(g)≤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).