Asymptotic normality of embedding distributions of some families of graphs

TL;DR

This paper investigates the asymptotic behavior of embedding distributions in certain graph families, demonstrating they often tend toward a normal distribution, and introduces new methods for establishing this property.

Contribution

It develops new tools and frameworks to prove the asymptotic normality of embedding distributions in various graph families, expanding understanding in topological graph theory.

Findings

Proves asymptotic normality for several graph families

Introduces adaptable tools for analyzing embedding distributions

Raises open questions and conjectures in the field

Abstract

Computing the embedding distribution of a given graph is a fundamental question in topological graph theory. In this article, we extend our viewpoint to a sequence of graphs and consider their asymptotic embedding distributions, which are often the normal distribution. We establish the asymptotic normality of several families of graphs by developing adapted tools and frameworks. We expect that these tools and frameworks can be used on other families of graphs to establish the asymptotic normality of their embedding distributions. Several open questions and conjectures are also raised in our investigation.