Genus Bounds for Harmonic Group Actions on Finite Graphs

This paper develops graph analogs of the genus bounds for the maximal size of an auto-morphism 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).