Related papers: Proper caterpillars are distinguished by their sym…
Stanley's Tree Isomorphism Conjecture posits that the chromatic symmetric function can distinguish non-isomorphic trees. While already established for caterpillars and other subclasses of trees, we prove the conjecture's validity for a new…
A well-known open problem in graph theory asks whether Stanley's chromatic symmetric function, a generalization of the chromatic polynomial of a graph, distinguishes between any two non-isomorphic trees. Previous work has proven the…
Stanley asked whether a tree is determined up to isomorphism by its chromatic symmetric function. We approach Stanley's problem by studying the relationship between the chromatic symmetric function and other invariants. First, we prove…
Stanley introduced the concept of chromatic symmetric functions of graphs which extends and refines the notion of chromatic polynomials of graphs, and asked whether trees are determined up to isomorphism by their chromatic symmetric…
In this paper, we propose an algebraic approach to determine whether two non-isomorphic caterpillar trees can have the same symmetric function generalization of the chromatic polynomial. On the set of all composition on integers, we…
This paper contributes to a programme initiated by the first author: `How much information about a graph is revealed in its Potts partition function?'. We show that the $W$-polynomial distinguishes non-isomorphic weighted trees of a…
Let $T$ be an unrooted tree. The \emph{chromatic symmetric function} $X_T$, introduced by Stanley, is a sum of monomial symmetric functions corresponding to proper colorings of $T$. The \emph{subtree polynomial} $S_T$, first considered…
A famous conjecture of Stanley states that his chromatic symmetric function distinguishes trees. As a quasisymmetric analogue, we conjecture that the chromatic quasisymmetric function of Shareshian and Wachs and of Ellzey distinguishes…
In 1995, Stanley introduced the well-known chromatic symmetric function $X_{G}(x_{1},x_{2},\ldots)$ of a graph $G$. It is a sum of monomial symmetric functions such that for each vertex coloring of $G$ there is exactly one of these…
One of the major outstanding conjectures in the study of chromatic symmetric functions (CSF's) states that trees are uniquely determined by their CSF's. Though verified on graphs of order up to twenty-nine, this result has been proved only…
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 $U_k$-polynomial, which is a…
Stanley introduced the chromatic symmetric function of a simple graph, which is a generalization of a chromatic polynomial. This is expressed in terms of the integer points of the complements of the corresponding graphic arrangement.…
Richard P. Stanley defined the chromatic symmetric function of a simple graph and has conjectured that every tree is determined by its chromatic symmetric function. Recently, Takahiro Hasebe and the author proved that the order…
Richard P. Stanley conjectured that finite trees can be distinguished by their chromatic symmetric functions. In this paper, we prove an analogous statement for posets: Finite rooted trees can be distinguished by their order quasisymmetric…
It is a long-standing question of Stanley whether or not the chromatic symmetric function (CSF) distinguishes unrooted trees. Previously, the best computational result, due to Russell, proved that it distinguishes all trees with at most…
For a tree $T$, consider its smallest subtree $T^{\circ}$ containing all vertices of degree at least $3$. Then the remaining edges of $T$ lie on disjoint paths each with one endpoint on $T^{\circ}$. We show that the chromatic symmetric…
R.P. Stanley defined a invariant for graphs called the chromatic symmetric function and conjectured it is complete invariant for trees. Miezaki et al. generalised the chromatic symmetric function and defined the Kneser chromatic functions…
Stanley [9] introduced the chromatic symmetric function ${\bf X}_G$ associated to a simple graph $G$ as a generalization of the chromatic polynomial of $G$. In this paper we present a novel technique to write ${\bf X}_G$ as a linear…
A caterpillar tree is a connected, acyclic, graph in which all vertices are either a member of a central path, or joined to that central path by a single edge. In other words, caterpillar trees are the class of trees which become path…
Stanley associated with a graph G a symmetric function X_G which reduces to G's chromatic polynomial under a certain specialization of variables. He then proved various theorems generalizing results about the chromatic polynomial, as well…