Counting One-Vertex Maps

TL;DR

This paper calculates the number of unique one-vertex maps with various configurations, considering different graph types such as signed, unsigned, directed, or undirected, for any valence d.

Contribution

It provides a comprehensive enumeration of one-vertex maps across multiple graph types, extending previous work to include signed and directed cases.

Findings

Derived formulas for counting maps with a single vertex and given valence.

Enumerated maps for both signed and unsigned, directed and undirected graphs.

Established a general method applicable to various map types.

Abstract

The number of distinct maps (pre-maps) with a single vertex and valence $d$ is computed for any value of $d$. The types of maps (pre-maps) that we consider depend on whether the underlaying graph (pre-graph) is signed or unsigned and directed or undirected.