The μ-polynomials of graph associahedra.
Ávila Ramírez, Nicolás
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We study two polynomials associated to a graph G that are of interest in the recent literature. The first one is the h-polynomial of the graph-associahedron of G defined by Carr and Devadoss. The second one is the μ-polynomial recently defined by González D’León and Wachs, which in the case of trees the authors conjecture that is up to sign equal to its h-polynomial. We prove a more general relation between the h- and the μ-polynomial of G which in the special case of trees proves González D’León - Wachs’ conjecture. We give a new description of the μ-polynomials in terms of a family of forests that we call μ-forests. As applications of the tools developed, we compute the μ- polynomials of the families of cycle and kite-like graphs. These are related to the Narayana polynomials of type A and B. We also show that these families of polynomials are realrooted and form interlacing sequences giving support and extending previous conjectures to a general conjecture on real-rootedness and the interlacing property of the h- and the μ-polynomials of an arbitrary graph G.