Supersolvability and Freeness for $\psi$-Graphical Arrangements

Abstract

Let $G$ be a simple graph on the vertex set $\{v_1,\dots ,v_n\}$ with edge set $E$. Let $K$ be a field. The graphical arrangement ${\mathcal {A}}_G$ in $K^n$ is the arrangement $x_i\!-\!x_j\!=\!0, v_iv_j \in E$. An arrangement ${\mathcal {A}}$ is supersolvable if the intersection lattice $L(c({\mathcal {A}}))$ of the cone $c({\mathcal {A}})$ contains a maximal chain of modular elements. The second author has shown that a graphical arrangement ${\mathcal {A}}_G$ is supersolvable if and only if $G$ is a chordal graph. He later considered a generalization of graphical arrangements which are called $\psi$-graphical arrangements. He conjectured a characterization of the supersolvability and freeness (in the sense of Terao) of a $\psi$-graphical arrangement. We provide a proof of the first conjecture and state some conditions on free $\psi$-graphical arrangements.