Harmonic morphisms and hyperelliptic graphs

Matthew Baker; Serguei Norine

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arXiv · 0707.1309

Harmonic morphisms and hyperelliptic graphs

Matthew Baker·Serguei Norine

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Abstract

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 2-edge-connected graph G which is not a cycle, there is at most one involution $ι$ on G for which the quotient $G/ι$ 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 non-constant harmonic morphism to the graph B_2 consisting of 2 vertices connected by 2 edges. Finally, we use the Riemann-Hurwitz formula and our results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points.

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BibTeX RIS

Matthew Baker, Serguei Norine. 2007-07-18. Harmonic morphisms and hyperelliptic graphs. https://arxiv.org/abs/0707.1309

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