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# Arkiv för Matematik

## Volume 56 (2018)

### Number 2

### Laplacian simplices associated to digraphs

Pages: 243 – 264

DOI: http://dx.doi.org/10.4310/ARKIV.2018.v56.n2.a3

#### Authors

#### 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.

#### Keywords

lattice polytope, Laplacian simplex, digraph, spanning tree, matrixtree theorem

#### 2010 Mathematics Subject Classification

Primary 52B20. Secondary 05C20.

Received 11 October 2017

Received revised 15 March 2018