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, $h^*$-polynomial, and 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.