On distinguishing trees by their chromatic symmetric functions

Abstract

Let T be an unrooted tree. The chromatic symmetric function X T , introduced by Stanley, is a sum of monomial symmetric functions corresponding to proper colorings of T. The subtree polynomial S T , first considered under a different name by Chaudhary and Gordon, is the bivariate generating function for subtrees of T by their numbers of edges and leaves. We prove that S T = 〈 Φ , X T 〉 , where 〈 ⋅ , ⋅ 〉 is the Hall inner product on symmetric functions and Φ is a certain symmetric function that does not depend on T. 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 ( spiders and some caterpillars), and one family of unicyclic graphs ( squids) whose members are determined completely by their chromatic symmetric functions.