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arXiv · 1511.04853

Vertex-weighted graphs and freeness of $ ψ$-graphical arrangements

Daisuke Suyama·Shuhei Tsujie
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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 $ ψ$-graphical arrangement $ \mathcal{A}_{G, ψ} $ as a generalization of graphical arrangements. Lili Mu and Stanley characterized the supersolvability of the $ ψ$-graphical arrangements and conjectured that the freeness and the supersolvability of $ ψ$-graphical arrangements are equivalent. In this paper, we will prove the conjecture.

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Daisuke Suyama, Shuhei Tsujie. 2017-04-20. Vertex-weighted graphs and freeness of $ ψ$-graphical arrangements. https://doi.org/10.1007/s00454-018-9984-1

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