Enumeration of Tree-like Maps with Arbitrary Number of Vertices

TL;DR

This paper develops generating series for tree-like graph embeddings of any size across different genera, extending existing formulas and techniques to generalize the Harer-Zagier formula for moduli space Euler characteristics.

Contribution

It introduces a comprehensive method to enumerate tree-like maps with any number of vertices, generalizing key formulas in topological graph theory.

Findings

Provides generating series for tree-like maps of arbitrary size and genus.

Extends Chan's techniques to broader classes of graphs.

Generalizes the Harer-Zagier formula to multiple vertices.

Abstract

This paper provides the generating series for the embedding of tree-like graphs of arbitrary number of vertices, accourding to their genus. It applies and extends the techniques of Chan, where it was used to give an alternate proof of the Goulden and Slofstra formula. Furthermore, this greatly generalizes the famous Harer-Zagier formula, which computes the Euler characteristic of the moduli space of curves, and is equivalent to the computation of one vertex maps.