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

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.