Harmonic morphisms and hyperelliptic graphs
| dc.creator | Baker, Matthew | |
| dc.creator | Norine, Serguei | |
| dc.date | 2007-07-09 | |
| dc.date | 2007-07-18 | |
| dc.date.accessioned | 2026-07-07T08:18:41Z | |
| dc.date.available | 2026-07-07T08:18:41Z | |
| dc.description | 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. | |
| dc.description | 29 pages, 2 figures. v2: Added Corollary 4.25, Remark 4.26, and Corollary 5.21, and made minor stylistic changes | |
| dc.identifier | https://arxiv.org/abs/0707.1309 | |
| dc.identifier | http://arxiv.org/abs/0707.1309 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/134519 | |
| dc.subject | Combinatorics | |
| dc.subject | Algebraic Geometry | |
| dc.title | Harmonic morphisms and hyperelliptic graphs | |
| dc.type | text |