Arkiv för Matematik

  • Ark. Mat.
  • Volume 56, Number 2 (2018), 243-264.

Laplacian simplices associated to digraphs

Gabriele Balletti, Takayuki Hibi, Marie Meyer, and Akiyoshi Tsuchiya

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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.

Article information

Ark. Mat., Volume 56, Number 2 (2018), 243-264.

Received: 11 October 2017
Revised: 15 March 2018
First available in Project Euclid: 19 June 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]
Secondary: 05C20: Directed graphs (digraphs), tournaments

lattice polytope Laplacian simplex digraph spanning tree matrixtree theorem


Balletti, Gabriele; Hibi, Takayuki; Meyer, Marie; Tsuchiya, Akiyoshi. Laplacian simplices associated to digraphs. Ark. Mat. 56 (2018), no. 2, 243--264. doi:10.4310/ARKIV.2018.v56.n2.a3.

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