Laplacian simplices associated to digraphs

Gabriele Balletti Stockholm University Takayuki Hibi Osaka University Marie Meyer University of Kentucky Akiyoshi Tsuchiya Osaka University

Combinatorics mathscidoc:1912.43014

Arkiv for Matematik, 56, (2), 243 – 264, 2018
We associate to a finite digraph D a lattice polytope PD 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 PD equals the complexity of D, and PD 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 PD 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.
lattice polytope, Laplacian simplex, digraph, spanning tree, matrixtree theorem
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  title={Laplacian simplices associated to digraphs},
  author={Gabriele Balletti, Takayuki Hibi, Marie Meyer, and Akiyoshi Tsuchiya},
  booktitle={Arkiv for Matematik},
  pages={243 – 264},
Gabriele Balletti, Takayuki Hibi, Marie Meyer, and Akiyoshi Tsuchiya. Laplacian simplices associated to digraphs. 2018. Vol. 56. In Arkiv for Matematik. pp.243 – 264.
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