Combinatorics of Even-Valent Graphs on Riemann Surfaces

TL;DR

This paper develops a systematic framework using random matrix theory and orthogonal polynomials to derive explicit formulas for counting even-valent graphs embedded on Riemann surfaces of any genus, extending known results to higher genera.

Contribution

It provides a method to explicitly compute the coefficients in structural formulas for graph counts on Riemann surfaces of arbitrary genus, completing fixed genus combinatorics for these graphs.

Findings

Explicit formulas for graph counts on surfaces of genus 2, 3, and 4.

Framework extends to all genera g ≥ 5 with more computation.

Recovers known results for planar and torus embeddings.

Abstract

Using connections to random matrix theory and orthogonal polynomials, we develop a framework for obtaining explicit closed-form formulae for the number, $\mathscr{N}_{g}(2\nu,j)$, of connected $2\nu$-valent labeled graphs with $j$ vertices that can be embedded on a compact Riemann surface of minimal genus $g$. We also derive formulae for their two-legged counterparts $\mathcal{N}_{g}(2\nu,j)$. Our method recovers the known explicit results for graphs embedded on the plane and the torus, and extends them to all genera $g \geq 2$. In earlier work, Ercolani, Lega, and Tippings (2023) showed that $\mathscr{N}_{g}(2\nu,j)$ and $\mathcal{N}_{g}(2\nu,j)$ admit structural expressions as linear combinations of, respectively, $3g-2$ and $3g$ Gauss hypergeometric functions ${}_2F_1$, but with coefficients left undetermined. The framework developed here provides a systematic procedure to compute these coefficients, thereby turning the structural expressions into fully explicit formulae for $\mathscr{N}_{g}(2\nu,j)$ and $\mathcal{N}_{g}(2\nu,j)$ as functions of both $j$ and $\nu$. Detailed results are given for $g=2,3,$ and $4$, and the framework extends naturally to all $g \geq 5$ with increasing computational effort. This closes the fixed genus combinatorics for even-valent graphs.