Vertex-weighted graphs and freeness of $ \psi $-graphical arrangements

TL;DR

This paper proves a conjecture that for $ ext{psi} $-graphical arrangements, freeness and supersolvability are equivalent, extending known results from classical graphical arrangements to this generalized setting.

Contribution

It establishes the equivalence of freeness and supersolvability for $ ext{psi} $-graphical arrangements, confirming a conjecture by Mu and Stanley.

Findings

Freeness and supersolvability are equivalent for $ \psi $-graphical arrangements.

The conjecture by Mu and Stanley is proven.

Generalizes classical results on graphical arrangements.

Abstract

Let $ G $ be a simple graph of $ \ell $ vertices $ \{1, \dots, \ell \} $ with edge set $ E_{G} $. The graphical arrangement $ \mathcal{A}_{G} $ consists of hyperplanes $ \{x_{i}-x_{j}=0\} $, where $ \{i, j \} \in E_{G} $. It is well known that three properties, chordality of $ G $, supersolvability of $ \mathcal{A}_{G} $, and freeness of $ \mathcal{A}_{G} $ are equivalent. Recently, Richard P. Stanley introduced $ \psi $-graphical arrangement $ \mathcal{A}_{G, \psi} $ as a generalization of graphical arrangements. Lili Mu and Stanley characterized the supersolvability of the $ \psi $-graphical arrangements and conjectured that the freeness and the supersolvability of $ \psi $-graphical arrangements are equivalent. In this paper, we will prove the conjecture.