Résumé : The U-polynomial of a graph was introduced by Noble and Welsh as a generalization of some invariants coming from Knot theory. It also generalizes the chromatic symmetric function of Stanley. In this talk, we will consider the problem of whether there exist non-isomorphic trees with the same U-polynomial (or,equivalently, with the same chromatic symmetric function). We will survey what is know about the U-polynomial and this problem. In particular, we will show how to recover some classic invariants from the U-polynomial and we exhibit several subclasses of trees for which a solution of this problem is known. FInally, we construct some non-isomorphic trees with "almost" the same U-polynomial, based on solutions of an old problem in Number theory due to Prouhet-Tarry-Escott.
[arXiv]
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