Séminaire Lotharingien de Combinatoire, 95B.92 (2026), 12 pp.
Aram Bingham, Lisa Johnston, Colin Lawson, Rosa Orellana, Jianping Pan and Chelsea Sato
The Chromatic Symmetric Function for Unicyclic Graphs
Abstract.
Motivated by the question of which structural properties of a graph can be recovered from the chromatic symmetric function (CSF), we study the CSF of connected unicyclic graphs.
While it is known that there can be non-isomorphic unicyclic graphs with the same CSF, we find that in many cases we can recover data such as the number of leaves, number of internal edges, cycle size, and number of attached non-trivial trees, by extending known results for trees to unicyclic graphs.
These results are obtained by analyzing the CSF of a connected unicyclic graph in the star basis using the deletion-near-contraction (DNC) relation developed by Aliste-Prieto, Orellana and Zamora, and computing the ``leading'' partition, as well as coefficients indexed by hook partitions.
We also show that the cycle sizes of bicyclic graphs can be recovered, and we obtain explicit formulas for star-expansions of specific families of graphs.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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