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Universal graph series and vertex-weighted version of chromatic symmetric function

Yosuke Sato

TL;DR

The paper develops a unified framework to generalize the chromatic symmetric function through universal graphs and vertex-weighted graphs. It shows that when $H$ is universal, $X_H(\\bullet)$ (and its vertex-weighted analog) are complete invariants for finite graphs, and extends these invariants to DAGs and posets. A power-sum expansion for $X_{K_{\\mathbb{N},k}}(G,w)$ is derived via weight-homomorphisms and admissible $\\mathcal{P}^{(k)}$-classes, enabling refined discriminative power. The results link graph invariants to hyperplane arrangements through intersection posets, offering broad implications for combinatorial structure classification.

Abstract

We focus on two specific generalizations of the chromatic symmetric function: one involving universal graphs and the other concerning vertex-weighted graphs. In this paper, we introduce a unified generalization that incorporates both approaches and demonstrate that the resulting new invariants inherit characteristics from each, particularly the properties of complete invariants. Additionally, we construct complete invariants for directed acyclic graphs (DAGs) and partially ordered sets (posets). As a corollary, these invariants can distinguish hyperplane arrangements that are distinguishable by their intersection posets.

Universal graph series and vertex-weighted version of chromatic symmetric function

TL;DR

The paper develops a unified framework to generalize the chromatic symmetric function through universal graphs and vertex-weighted graphs. It shows that when is universal, (and its vertex-weighted analog) are complete invariants for finite graphs, and extends these invariants to DAGs and posets. A power-sum expansion for is derived via weight-homomorphisms and admissible -classes, enabling refined discriminative power. The results link graph invariants to hyperplane arrangements through intersection posets, offering broad implications for combinatorial structure classification.

Abstract

We focus on two specific generalizations of the chromatic symmetric function: one involving universal graphs and the other concerning vertex-weighted graphs. In this paper, we introduce a unified generalization that incorporates both approaches and demonstrate that the resulting new invariants inherit characteristics from each, particularly the properties of complete invariants. Additionally, we construct complete invariants for directed acyclic graphs (DAGs) and partially ordered sets (posets). As a corollary, these invariants can distinguish hyperplane arrangements that are distinguishable by their intersection posets.

Paper Structure

This paper contains 7 sections, 13 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Universal graph series and chromatic symmetric function
  4. Extension of the chromatic symmetric function to the vertex-weighted graphs.
  5. Extending $X_H(G)$ to Vertex-Weighted Graphs
  6. Power sum expansion of $X_{K_{\mathbb{N},k}}(G,w)$
  7. Construction of complete invariants for DAGs and posets

Key Result

Theorem 1.3

where, for each subset $S\subseteq E$, $\lambda_S$ denotes the partition of $|V(G)|$ given by the vertex counts in each connected component. For a partition $\lambda=(\lambda_1,\cdots,\lambda_l)$, we define

Figures (1)

  • Figure 1:

Theorems & Definitions (33)

  • Definition 1.1: Sta95
  • Definition 1.2
  • Theorem 1.3
  • Definition 2.1: MMNST,Rado
  • Definition 2.2
  • Theorem 2.3: MMNST
  • Theorem 2.4: MMNST, Theorem 1.9
  • Definition 2.5: CS
  • Proposition 2.6: CS, Lemma 2
  • Theorem 2.7: CS, Lemma 3
  • ...and 23 more