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

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.

Marked graphs and the chromatic symmetric function

“…Example 12. The following two trees were given in [18] as an example of two non-isomorphic weighted graphs with the same W polynomial (for the general construction of paths with the same Wpolynomial see [1,3,6]). Note that these two trees can be thought as marked trees with marks (w, 0) and as such they have different M -polynomial.…”

Plethysms of Chromatic and Tutte Symmetric Functions

“…The Tutte (or bad-colouring) symmetric function XB (G,w) is defined as a symmetric function with coefficient ring C[t] as [4,35]…”

“…where the sum ranges over all colourings of G (not just the proper ones), with e(κ) equal to the number of edges of G whose endpoints receive the same colour from κ. From [4] we also have…”

“…It may be shown that this function satisfies the deletion-contraction relation [4] XB (G,w) = XB (G\e,w) + tXB (G/e,w/e) .…”

“…We also consider plethysms involving the Tutte symmetric function, also introduced by Stanley [36] and extended to vertex-weighted graphs by Aliste-Prieto, Zamora, and the authors [4] as a symmetric function with coefficient ring C[t] defined by…”