Related papers: Homogeneous Sets in Graphs and a Chromatic Multisy…
A well-known result of Stanley's shows that given a graph $G$ with chromatic symmetric function expanded into the basis of elementary symmetric functions as $X_G = \sum c_{\lambda}e_{\lambda}$, the sum of the coefficients $c_{\lambda}$ for…
We introduce path-conjoined graphs defined for two rooted graphs by joining their roots with a path, and investigate the chromatic symmetric functions of its two generalizations: spider-conjoined graphs and chain-conjoined graphs. By using…
In this paper, we introduce and study two variants of the chromatic quasisymmetric function of a graph: the total chromatic quasisymmetric function via vertex labeling and via acyclic orientations. The original definition of the chromatic…
In 1995, Richard Stanley introduced the chromatic symmetric function $X_G$ of a graph $G$ and proved that, when written in terms of the elementary symmetric functions, it reveals the number of acyclic orientations of $G$ with a given number…
Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize the chromatic polynomial and are related to Hessenberg varieties and diagonal harmonics. Motivated by the Stanley--Stembridge…
In this paper we introduce the notion of $\Sigma$-colouring of a graph $G$: For given subsets $\Sigma(v)$ of neighbours of $v$, for every $v\in V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that…
We prove that the coefficient of the star $\mathfrak{st}_{21^{n-2}}$ in the chromatic symmetric function $X_G$ determines whether a connected graph $G$ is $2$-connected. We also prove new linear relations on other star coefficients of…
By using level one polynomial representations of affine Hecke algebras of type $A$, we obtain a $(q,t)$-analogue of the chromatic symmetric functions of unit interval graphs which generalizes Syu Kato's formula for the chromatic symmetric…
We consider a refinement of the partition function of graph homomorphisms and present a quasi-polynomial algorithm to compute it in a certain domain. As a corollary, we obtain quasi-polynomial algorithms for computing partition functions…
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…
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…
Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infnitely many variables. The ordinary chromatic polynomial is a specialization…
In this paper, we investigate the strength of chromatic symmetric homology as a graph invariant. Chromatic symmetric homology is a lift of the chromatic symmetric function for graphs to a homological setting, and its Frobenius…
In Chapter 2 we study the path-cycle symmetric function of a digraph, a symmetric function generalization of Chung and Graham's cover polynomial. Most of this material appears in either Advances in Math. 118 (1996), 71-98 or J. Algebraic…
We define an algorithm k which takes a connected graph G on a totally ordered vertex set and returns an increasing tree R (which is not necessarily a subtree of G). We characterize the set of graphs G such that k(G)=R. Because this set has…
We introduce a quasisymmetric refinement of Stanley's chromatic symmetric function. We derive refinements of both Gasharov's Schur-basis expansion of the chromatic symmetric function and Chow's expansion in Gessel's basis of fundamental…
We investigate the problem of when a chromatic quasisymmetric function (CQF) $X_G(x;q)$ of a graph $G$ is in fact symmetric. We first prove the remarkable fact that if a product of two quasisymmetric functions $f$ and $g$ in countably…
For $k\ge 1$, we consider interleaved $k$-tuple colorings of the nodes of a graph, that is, assignments of $k$ distinct natural numbers to each node in such a way that nodes that are connected by an edge receive numbers that are strictly…
We prove a new signed elementary symmetric function expansion of the chromatic quasisymmetric function of any natural unit interval graph. We then use a sign-reversing involution to prove a new combinatorial formula for K-chains, which are…
A MacMahon symmetric function is an invariant of the diagonal action of the symmetric group on power series in multiple alphabets of variables. We introduce an analogue of the chromatic symmetric function for vertex-weighted graphs, taking…