Modular relations of the Tutte symmetric function
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.
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Cite this version of the work
Logan Crew, Sophie Spirkl
(2022).
Modular relations of the Tutte symmetric function. UWSpace.
http://hdl.handle.net/10012/18592
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