Arkiv för Matematik

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

Laplacian simplices associated to digraphs

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.

Article information

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

Dates
Revised: 15 March 2018
First available in Project Euclid: 19 June 2019

https://projecteuclid.org/euclid.afm/1560968132

Digital Object Identifier
doi:10.4310/ARKIV.2018.v56.n2.a3

Mathematical Reviews number (MathSciNet)
MR3893773

Zentralblatt MATH identifier
1408.52021

Citation

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. https://projecteuclid.org/euclid.afm/1560968132