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Laplacian simplices associated to digraphs
Stockholm University, Faculty of Science, Department of Mathematics.
Number of Authors: 42018 (English)In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 56, no 2, p. 243-264Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2018. Vol. 56, no 2, p. 243-264
Keywords [en]
lattice polytope, Laplacian simplex, digraph, spanning tree, matrix-tree theorem
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-166636DOI: 10.4310/ARKIV.2018.v56.n2.a3ISI: 000457426300003OAI: oai:DiVA.org:su-166636DiVA, id: diva2:1296341
Available from: 2019-03-14 Created: 2019-03-14 Last updated: 2019-03-14Bibliographically approved

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