Supersolvability and Freeness for $$\psi $$ ψ -Graphical Arrangements

Abstract

Let $$G$$G be a simple graph on the vertex set $$\{v_1,\dots ,v_n\}$${v1,?,vn} with edge set $$E$$E. Let $$K$$K be a field. The graphical arrangement $${\mathcal {A}}_G$$AG in $$K^n$$Kn is the arrangement $$x_i\!-\!x_j\!=\!0, v_iv_j \in E$$xi-xj=0,vivj?E. An arrangement $${\mathcal {A}}$$A is supersolvable if the intersection lattice $$L(c({\mathcal {A}}))$$L(c(A)) of the cone $$c({\mathcal {A}})$$c(A) contains a maximal chain of modular elements. The second author has shown that a graphical arrangement $${\mathcal {A}}_G$$AG is supersolvable if and only if $$G$$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.