A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

TL;DR

This paper introduces a vertex-weighted version of the Tutte symmetric function, extending its properties, and presents methods for constructing nonisomorphic graphs with identical chromatic and Tutte symmetric functions.

Contribution

It develops a vertex-weighted Tutte symmetric function with deletion-contraction and spanning-tree expansions, and provides new techniques for constructing nonisomorphic graphs sharing the same symmetric functions.

Findings

Vertex-weighted XB admits deletion-contraction relation.

Spanning-tree and spanning-forest expansions are generalized.

Methods for constructing graphs with equal chromatic and Tutte symmetric functions.

Abstract

This paper has two main parts. First, we consider the Tutte symmetric function $XB$, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of $XB$ and show that this function admits a deletion-contraction relation. We also demonstrate that the vertex-weighted $XB$ admits spanning-tree and spanning-forest expansions generalizing those of the Tutte polynomial by connecting $XB$ to other graph functions. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples.