An Extension of Stanley's Symmetric Acyclicity Theorem to Signed Graphs

TL;DR

This paper extends Stanley's symmetric acyclicity theorem to signed graphs, providing a new basis that captures sink counts and generalizes to both signed and unsigned graphs.

Contribution

It introduces a generalized theorem for signed graphs and a new basis that detects sinks, expanding Stanley's original results to more complex graph structures.

Findings

Generalization of Stanley's theorem to signed graphs

Introduction of a new basis detecting sinks

Analogous results for unsigned graphs

Abstract

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 of sinks. In this paper, we generalize this result to signed graphs, that is, to graphs whose edges are labeled with $+$ or $-$ and whose colorings and orientations can interact with their signs. Additionally, we introduce a non-homogeneous basis which detects the number of sinks and which not only gives a Stanley-type result for signed graphs but gives an analogous result of this form for unsigned graphs as well.