Proper $q$-caterpillars are distinguished by their Chromatic Symmetric Functions

G. Arunkumar, Narayanan Narayanan, Raghavendra Rao B. V., Sagar S. Sawant

Published 2023-07-05Version 1

Stanley's Tree Isomorphism Conjecture posits that the chromatic symmetric function can distinguish non-isomorphic trees. While already established for caterpillars and other subclasses of trees, we prove the conjecture's validity for a class of trees that generalize proper caterpillars, thus confirming the conjecture for a broader class of trees. Additionally, we exhibit a new multiplication operation on the symmetric functions such that the Tutte symmetric function of join of graphs splits into the respective Tutte symmetric functions of the individual graphs. This finding sheds new light on the interplay between graph operations and symmetric functions.