[Paper Review] Harmonic morphisms and hyperelliptic graphs

Matthew Baker, Serguei Norine|ArXiv.org|Jul 9, 2007

Algebraic Geometry and Number Theory6 references10 citations

TL;DR

This paper introduces harmonic morphisms as a discrete analogue of holomorphic maps between Riemann surfaces, establishing a graph-theoretic Riemann-Hurwitz formula and characterizing hyperelliptic graphs. It proves that a 2-edge-connected graph has at most one hyperelliptic involution, and classifies all hyperelliptic graphs with no Weierstrass points as $B_n$, $B(l_1,l_2,l_3)$ with odd $l_i$, or $\Phi(l)$.

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.

Motivation & Objective

• To develop a graph-theoretic analogue of holomorphic maps between Riemann surfaces using harmonic morphisms.
• To formulate a Riemann-Hurwitz formula for graphs, relating genus, ramification, and degree of harmonic morphisms.
• To characterize hyperelliptic graphs—graphs admitting a degree-2 harmonic morphism to a tree—via multiple equivalent definitions.
• To classify all 2-edge-connected hyperelliptic graphs with no Weierstrass points.
• To establish connections between harmonic morphisms and combinatorial invariants such as the number of spanning trees.

Proposed method

• Define harmonic morphisms between multigraphs via horizontal conformality: (HC1) adjacent vertices map to adjacent or equal vertices; (HC2) preimage degrees are constant over neighbors.
• Introduce vertex ramification index $v_\phi(x)$ and multiplicity $m_\phi(x)$ to define the degree of a harmonic morphism.
• Derive a Riemann-Hurwitz formula: $\sum_{x \in V(G)} v_\phi(x) = 2g(G) + 2 - 2g(G') \cdot \deg(\phi)$, analogous to the classical formula.
• Use functorial maps on Jacobians and harmonic 1-forms induced by harmonic morphisms to study linear systems on graphs.
• Apply the Riemann-Hurwitz formula to classify hyperelliptic graphs with no Weierstrass points by analyzing vertex degrees and preimage structures.
• Use divisor theory on graphs, including rank $r(D)$ and linear equivalence, to verify the absence of Weierstrass points in candidate graphs.

Experimental results

Research questions

• RQ1 What is the correct graph-theoretic analogue of a holomorphic map between Riemann surfaces?
• RQ2 How can a Riemann-Hurwitz formula be formulated in the context of finite graphs and harmonic morphisms?
• RQ3 What are the equivalent characterizations of a hyperelliptic graph in the graph-theoretic setting?
• RQ4 When does a 2-edge-connected graph admit at most one hyperelliptic involution?
• RQ5 Which hyperelliptic graphs have no Weierstrass points, and how can they be completely classified?

Key findings

• For a 2-edge-connected graph $G$ that is not a cycle, there is at most one involution $\iota$ such that the quotient $G/\iota$ is a tree.
• 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 $B_2$ (two vertices with two edges).
• All 2-edge-connected hyperelliptic graphs with no Weierstrass points are isomorphic to $B_n$ for $n \geq 3$, $B(l_1,l_2,l_3)$ with odd $l_i \geq 1$, or $\Phi(l)$ for $l \geq 1$.
• Graphs $B_n$ have no Weierstrass points because $r((n-1)(x) - (y)) = -1$, implying $r((n-1)(x)) = 0$.
• The graphs $B(l_1,l_2,l_3)$ with odd $l_i$ have no Weierstrass points because the hyperelliptic involution has no fixed points.
• The graphs $\Phi(l)$ have no Weierstrass points because $r(3(x_i)) = 0$ for all vertices $x_i$, verified via divisor linear equivalence and chip-firing arguments.