Maximal harmonic group actions on finite graphs

This paper studies groups of maximal size acting harmonically on a finite graph. Our main result states that these maximal graph groups are exactly the finite quotients of the modular group $\mathrm{\Gamma }=<x,y\mathrm{\mid }{x}^2=y^3=1>$ of size at least 6. This characterization may be viewed as a discrete analogue of the description of Hurwitz groups as finite quotients of the $(2,3,7)$-triangle group in the context of holomorphic group actions on Riemann surfaces. In fact, as an immediate consequence of our result, every Hurwitz group is a maximal graph group, and the final section of the paper establishes a direct connection between maximal graphs and Hurwitz surfaces via the theory of combinatorial maps.