The chromatic symmetric function of graphs glued at a single vertex

TL;DR

This paper develops a matrix-analytic framework for the chromatic symmetric function of graphs formed by gluing at a single vertex. By linking forest triples with Hikita’s tableau probabilities, it ties two combinatorial descriptions of $X_G(\bm x)$ and proves new $e$-positivity results for graphs built from unit interval graphs and cycles, including cases arising from gluing the first and last vertices of such sequences. It shows that the chromatic symmetric function of a glued graph $G+H$ can be computed as a product of information-carrying matrices $F_G$ and $F_H$, and introduces a tableau-based matrix $T_G(q)$ that coincides with $F_G(q)$ for NUIGs, yielding both $e$-positivity and structural closure under gluing. The trace formula $X_{G^ullet}(m x)=\mathrm{trace}(F_G)$ provides a practical tool for analyzing graphs formed by identifying specified vertices, with implications for proper circular-arc graphs and Ellzey-type conjectures. Overall, the work unifies combinatorial, tableau-based, and matrix approaches to advance $e$-positivity results in graph chromatic symmetric functions and their graph-operations.

Abstract

We describe how the chromatic symmetric function of two graphs glued at a single vertex can be expressed as a matrix multiplication using certain information of the two individual graphs. We then prove new $e$-positivity results by using a connection between forest triples, defined by the first author, and Hikita's probabilities associated to standard Young tableaux. Specifically, we prove that gluing a sequence of unit interval graphs and cycles results in an $e$-positive graph. We also prove $e$-positivity for a graph obtained by gluing the first and last vertices of such a sequence. This generalizes $e$-positivity of cycle-chord graphs and supports Ellzey's conjectured $e$-positivity for proper circular arc digraphs.

Figures (9)

• Figure 1: Graphs $G$ and $H$ glued at a single vertex.
• Figure 2: The bowtie graph $G$ and the chromatic quasisymmetric function $X_G(\bm x;q)$.
• Figure 3: The directed cycle $\vec{C}_6$ and the chromatic quasisymmetric function $X_{\vec{C}_6}(\bm x;q)$.
• Figure 4: The bowtie graph $G$ and the forest triples of $G$ of type $32$.
• Figure 5: The bowtie graph $G$ and the standard Young tableaux in $\text{SYT}(G)$.

Theorems & Definitions (56)

• Theorem 2.1
• Conjecture 2.2
• Example 2.3
• Example 2.4
• Example 2.5
• Conjecture 2.6
• Example 2.7
• Definition 2.8
• Definition 2.9
• Theorem 2.10