Abstract
We associate to a finite digraph $D$ a lattice polytope $P_D$ whose vertices are the rows of the Laplacian matrix of $D$. This generalizes a construction introduced by Braun and the third author. As a consequence of the Matrix-Tree Theorem, we show that the normalized volume of $P_D$ equals the complexity of $D$, and $P_D$ contains the origin in its relative interior if and only if $D$ is strongly connected. Interesting connections with other families of simplices are established and then used to describe reflexivity, the $h^{*}$-polynomial, and the integer decomposition property of $P_D$ in these cases. We extend Braun and Meyer’s study of cycles by considering cycle digraphs. In this setting, we characterize reflexivity and show there are only four non-trivial reflexive Laplacian simplices having the integer decomposition property.
Citation
Gabriele Balletti. Takayuki Hibi. Marie Meyer. Akiyoshi Tsuchiya. "Laplacian simplices associated to digraphs." Ark. Mat. 56 (2) 243 - 264, October 2018. https://doi.org/10.4310/ARKIV.2018.v56.n2.a3
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