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

TL;DR

This paper introduces a unified generalization of the chromatic symmetric function that combines universal graphs and vertex-weighted graphs, creating complete invariants for DAGs and posets, and distinguishing hyperplane arrangements by their intersection posets.

Contribution

It presents a new unified invariant for graphs and posets that inherits properties from existing invariants and can distinguish complex combinatorial structures.

Findings

Introduced a unified generalization of chromatic symmetric functions.

Constructed complete invariants for DAGs and posets.

Showed invariants distinguish hyperplane arrangements by intersection posets.

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.