Laplacian Simplices Associated to Digraphs
Gabriele Balletti, Takayuki Hibi, Marie Meyer, and Akiyoshi Tsuchiya

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 a lattice polytope whose vertices are the rows of the Laplacian matrix of . 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 equals the complexity of , and contains the origin in its relative interior if and only if is strongly connected. Interesting connections with other families of simplices are established and then used to describe reflexivity, -polynomial, and integer decomposition property of 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.
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