Chromatic bases for symmetric functions

TL;DR

This paper introduces new bases for the algebra of symmetric functions generated by chromatic symmetric functions of specific graph classes, providing explicit formulas for several common graph types.

Contribution

It establishes that chromatic symmetric functions of connected graphs with increasing vertices form a generating set for the algebra, including explicit formulas for key graph families.

Findings

Chromatic symmetric functions of certain graphs generate the algebra of symmetric functions.

Explicit expressions are derived for complete, star, path, and cycle graphs.

The set of all such chromatic symmetric functions forms a basis for the algebra.

Abstract

In this note we obtain numerous new bases for the algebra of symmetric functions whose generators are chromatic symmetric functions. More precisely, if $\{ G_ k \}_{k\geq 1}$ is a set of connected graphs such that $G_k$ has $k$ vertices for each $k$, then the set of all chromatic symmetric functions $\{ X_{G_ k} \}_{k\geq 1}$ generates the algebra of symmetric functions. We also obtain explicit expressions for the generators arising from complete graphs, star graphs, path graphs and cycle graphs.