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On distinguishing trees by their chromatic symmetric functions

Abstract

Let TT be an unrooted tree. The \emph{chromatic symmetric function} XTX_T, introduced by Stanley, is a sum of monomial symmetric functions corresponding to proper colorings of TT. The \emph{subtree polynomial} STS_T, first considered under a different name by Chaudhary and Gordon, is the bivariate generating function for subtrees of TT by their numbers of edges and leaves. We prove that ST=S_T = , where is the Hall inner product on symmetric functions and Φ\Phi is a certain symmetric function that does not depend on TT. Thus the chromatic symmetric function is a stronger isomorphism invariant than the subtree polynomial. As a corollary, the path and degree sequences of a tree can be obtained from its chromatic symmetric function. As another application, we exhibit two infinite families of trees (\emph{spiders} and some \emph{caterpillars}), and one family of unicyclic graphs (\emph{squids}) whose members are determined completely by their chromatic symmetric functions.Comment: 16 pages, 3 figures. Added references [2], [13], and [15

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