Vertex-weighted graphs and freeness of $\psi$-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.

We added Chapter 6