A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function
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Date
2021-04-09
Authors
Aliste-Prieto, José
Crew, Logan
Spirkl, Sophie
Zamora, José
Advisor
Journal Title
Journal ISSN
Volume Title
Publisher
The Electronic Journal of Combinatorics
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 ofXB, show that this function admits a deletion-contraction relation, and show that it is equivalent to a number of other vertex-weighted graph functions, namely the W-polynomial, the polychromate, and the weighted (r, q)-chromatic function. We also demonstrate that the vertex-weighted X B admits spanning-tree and spanning-forest expansions generalizing those of the Tutte poly-nomial, and show that from this we may also derive a spanning-tree formula for the chromatic symmetric function. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples. In particular, we show that there are pairs of unweighted graphs of arbitrarily high girth with equal Tutte symmetric function, and arbitrarily large vertex-weighted trees with equal Tutte symmetric function
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Keywords
Tutte symmetric function, nonisomorphic