Digraph Laplacian spectra: Counterexamples to the 2005 polygon conjecture

Let $Λ_n$ be the set of all eigenvalues of standardized Laplacian matrices of order $n$. In 2005, a polygon $S_n\subseteqΛ_n$ was constructed, and it was conjectured that $Λ_n=S_n$. We observe that $S_n$ is a zonotope, which gives a short proof of $S_n\subseteqΛ_n$, and we describe $Λ_n$ as a union,