Harmonic morphisms and hyperelliptic graphs

By Matthew Baker, Sergey Norin. 2007

192 citations recorded by the source corpus.

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 $\iota$ 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 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.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/216bef3d938f33c50e5315246776def9d1615223

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Tropical hyperelliptic curves (2011, methodology): “. Whenis ametric graphhyperelliptic? Our goal in this section is to prove Theorem 3.13, which characterizes when a metric graph hyperelliptic. It is the metric analogue of one of the main theorems of [BN09]. We will ﬁrst prove the theorem for 2-edge-connected metric graphs in Theorem 3.2 and then pr”
Gonality of algebraic curves and graphs (2012, methodology): “ fact, in recent times a theory for divisors on graphs has been set-up and developed in a purely combinatorial way, revealing some remarkable analogies with the algebro-geometric case; see [BdlHN97], [BN09], [BN07] for example. One of the goals of this paper is to contribute to this developement; we”
Tropical independence II: The maximal rank conjecture for quadrics (2015, background): “The last statement, on surjectivity of µ 2 , also follows from Noether’s theorem, because trivalent, 3-edge connected graphs are never hyperelliptic [Baker and Norine 2009, Lemma 5.3].”
Specialization of linear systems from curves to graphs (2007, methodology): “These and other matters are discussed in [Baker and Norine 2007a].”
Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta (2013, unknown): “Since we will not use it elsewhere in the paper, we omit the proof of the following result (which is very similar to the proof of [8], Proposition 2.”
Lifting harmonic morphisms II: tropical curves and metrized complexes (2014, result): “The following proposition is a refinement of a result from [Chan 2013] on vertexweighted metric graphs (itself a strengthening of results from [Baker and Norine 2009]):”
Riemann–Roch theory for weighted graphs and tropical curves (2011, background): “udied independently of the tropical structures, 1 2 OMID AMINI AND LUCIA CAPORASO and also in this case the analogies with the classical theory of algebraic curves are quite compelling; see [BN1] and [BN2]. In this paper we are interested in divisor theory. For graphs and tropical curves, in the wei”