Harmonic Morphisms and Hyperelliptic Graphs

We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann–Hurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also discuss several equivalent formulations of the notion of a hyperelliptic graph, all motivated by the classical theory of Riemann surfaces. As an application of our results, we show that for a two-edge-connected graph G that is not a cycle, there is at most one involution $\iota$ on G for which the quotient $G/\iota$ is a tree. We also show that the number of spanning trees in a graph G is even if and only if G admits a nonconstant harmonic morphism to the graph B2 consisting of two vertices connected by two edges. Finally, we use the Riemann–Hurwitz formula and our results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points.