Structure of the chromatic polynomial

TL;DR

This paper applies topological data analysis and manifold learning to study the structure of chromatic polynomials of graphs, revealing insights into their properties and relation to random graph spaces.

Contribution

It introduces the use of TDA and manifold learning techniques to analyze chromatic polynomial data, providing new perspectives on their structure and invariants.

Findings

Filtered PCA and Ball Mapper reveal meaningful structures in chromatic polynomial data.

Chromatic polynomials cluster in ways related to graph invariants.

Results suggest connections between polynomial invariants and random graph models.

Abstract

Recently, big data techniques such as machine learning and topological data analysis have made their way to theoretical mathematics. Motivated by the recent work with polynomial invariants for knots, we use manifold learning and topological data analysis techniques to explore the structure and properties of the point cloud consisting of the chromatic polynomials of graphs up to 10 crossings. Although chromatic, as well as the Tutte polynomial fail to distinguish graphs, according to a conjecture by Bollobas, Pebody and Riordan they approximate the space of random graphs. In this work we compare structures in the chromatic data revealed using filtered PCA and Ball Mapper techniques, and relate them with a range of numerical invariants for graphs.