Proper caterpillars are distinguished by their symmetric chromatic function

This paper deals with the so-called Stanley conjecture, which asks whether they are non-isomorphic trees with the same symmetric function generalization of the chromatic polynomial. By establishing a correspondence between caterpillars trees and integer compositions, we prove that caterpillars in a large class (we call trees in this class proper) have the same symmetric chromatic function generalization of the chromatic polynomial if and only if they are isomorphic

Almost periodic structures and the semiconjugacy problem

The description of almost periodic or quasiperiodic structures has a long tradition in mathematical physics, in particular since the discovery of quasicrystals in the early 80's. Frequently, the modelling of such structures leads to different types of dynamical systems which include, depending on the concept of quasiperiodicity being considered, skew products over quasiperiodic or almost-periodic base flows, mathematical quasicrystals or maps of the real line with almost-periodic displacement. An important problem in this context is to know whether the considered system is semiconjugate to a rigid translation. We solve this question in a general setting that includes all the above-mentioned examples and also allows to treat scalar differential equations that are almost-periodic both in space and time. To that end, we study a certain class of flows that preserve a one-dimensional foliation and show that a semiconjugacy to a minimal translation flow exists if and only if a boundedness condition, concerning the distance of orbits of the flow to those of the translation, holds

On the simplicity of homeomorphism groups of a tilable lamination

We show that the identity component of the group of homeomorphisms that preserve all leaves of a R^d-tilable lamination is simple. Moreover, in the one dimensional case, we show that this group is uniformly perfect. We obtain a similar result for a dense subgroup of homeomorphisms.Comment: 14

On graphs with the same restricted U-polynomial and the U-polynomial for rooted graphs

In this abstract, we construct explicitly, for every k, pairs of non-isomorphic trees with the same restricted U-polynomial; by this we mean that the polynomials agree on terms with degree at most k. The construction is done purely in algebraic terms, after introducing and studying a generalization of the U-polynomial to rooted graphs.Peer ReviewedPostprint (author's final draft

On trees with the same restricted U-polynomial and the Prouhet–Tarry–Escott problem

This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same U-polynomial (or, equivalently, the same chromatic symmetric function). We consider the Uk-polynomial, which is a restricted version of U-polynomial, and construct, for any given kk, non-isomorphic trees with the same Uk-polynomial. These trees are constructed by encoding solutions of the Prouhet–Tarry–Escott problem. As a consequence, we find a new class of trees that are distinguished by the U-polynomial up to isomorphism.Peer ReviewedPostprint (author's final draft

A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

This paper has two main parts. First, we consider the Tutte symmetric function $XB$, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of $XB$ and show that this function admits a deletion-contraction relation. We also demonstrate that the vertex-weighted $XB$ admits spanning-tree and spanning-forest expansions generalizing those of the Tutte polynomial by connecting $XB$ to other graph functions. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples.Comment: 28 page

A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

This paper has two main parts. First, we consider the Tutte symmetric function XB, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version ofXB, show that this function admits a deletion-contraction relation, and show that it is equivalent to a number of other vertex-weighted graph functions, namely the W-polynomial, the polychromate, and the weighted (r, q)-chromatic function. We also demonstrate that the vertex-weighted X B admits spanning-tree and spanning-forest expansions generalizing those of the Tutte poly-nomial, and show that from this we may also derive a spanning-tree formula for the chromatic symmetric function. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples. In particular, we show that there are pairs of unweighted graphs of arbitrarily high girth with equal Tutte symmetric function, and arbitrarily large vertex-weighted trees with equal Tutte symmetric functionSupported by CONICYT FONDECYT REGULAR 1160975||National Science Foundation under Award No. DMS-1802201||Natural Sciences and Engineering Research Council of Canada (NSERC), (funding reference number RGPIN-2020-03912)