Modular relations of the Tutte symmetric function
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Date
2022-04
Authors
Crew, Logan
Spirkl, Sophie
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Abstract
For a graph G, its Tutte symmetric function XBG generalizes both the Tutte polynomial
TG and the chromatic symmetric function XG. We may also consider XB as a map from the
t-extended Hopf algebra 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 vanWilligenburg. 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.
Description
The final publication is available at Elsevier via https://doi.org/10.1016/j.jcta.2021.105572 © 2022. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Keywords
Tutte symmetric function, chromatic symmetric function, modular relation, hopf algebra, vertex-weighted graphs, algebraic combinatorics