Year: 2018
Type: article
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.
Source: Arkiv för matematik
Authors: Gabriele Balletti, Takayuki Hibi, Marie Meyer, Akiyoshi Tsuchiya
Language: Swedish
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W2761722424