Modular Relations of the Tutte Symmetric Function

TL;DR

This paper explores the algebraic structure of the Tutte symmetric function, revealing its kernel, modular relations, and connections to other graph invariants, thereby extending known results and providing new structural insights.

Contribution

It characterizes the kernel of the Tutte symmetric function, identifies key modular relations, and extends structural and expansion results from chromatic to Tutte symmetric functions.

Findings

Kernel generated by vertex-relabellings and modular relations

Generalization of the triangular modular relation of Orellana and Scott

Extension of chromatic symmetric function results to Tutte symmetric functions

Abstract

For a graph $G$, its Tutte symmetric function $XB_G$ generalizes both the Tutte polynomial $T_G$ and the chromatic symmetric function $X_G$. We may also consider $XB$ as a map from the $t$-extended Hopf algebra $\mathbb{G}[t]$ of labelled graphs to symmetric functions. We show that the kernel of $XB$ is generated by vertex-relabellings and a finite set of modular relations, in the same style as a recent analogous result by Penaguiao on the chromatic symmetric function $X$. In particular, we find one such relation that generalizes the well-known triangular modular relation of Orellana and Scott, and build upon this to give a modular relation of the Tutte symmetric function for any two-edge-connected graph that generalizes the $n$-cycle relation of Dahlberg and van Willigenburg. Additionally, we give a structural characterization of all local modular relations of the chromatic and Tutte symmetric functions, and prove that there is no single local modification that preserves either function on simple graphs. We also give an expansion relating $XB_G$ to $X_{G/S}$ as $S$ ranges over all subsets of $E(G)$, use this to extend results on the chromatic symmetric function to the Tutte symmetric function, and show that analogous formulas hold for a Tutte quasisymmetric function on digraphs.