Laplacian Simplices Associated to Digraphs

TL;DR

This paper introduces a new class of lattice polytopes derived from digraphs' Laplacian matrices, revealing their volume, interior points, and reflexivity properties, and extends previous work on cycles.

Contribution

It generalizes the construction of Laplacian simplices to digraphs, linking their geometric properties to graph-theoretic invariants and characterizing reflexivity and integer decomposition.

Findings

Normalized volume equals the graph's complexity

The polytope contains the origin iff the digraph is strongly connected

Only four non-trivial reflexive Laplacian simplices have the integer decomposition property

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.