  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 $M(g)$.
