Laplacian simplices

Benjamin Braun, Marie Meyer

Research output: Contribution to journalArticlepeer-review


This paper initiates the study of the Laplacian simplex TG obtained from a finite graph G by taking the convex hull of the columns of the Laplacian matrix for G. Basic properties of these simplices are established, and then a systematic investigation of TG for trees, cycles, and complete graphs is provided. Motivated by a conjecture of Hibi and Ohsugi, our investigation focuses on reflexivity, the integer decomposition property, and unimodality of Ehrhart h-vectors. We prove that if G is a tree, odd cycle, complete graph, or a whiskering of an even cycle, then TG is reflexive. We show that while TKn has the integer decomposition property, TCn for odd cycles does not. The Ehrhart h-vectors of TG for trees, odd cycles, and complete graphs are shown to be unimodal. As a special case it is shown that when n is an odd prime, the Ehrhart h-vector of TCn is given by (h0 ,…,hn−1 )=(1,…,1,n2−n+1,1,…,1). We also provide a combinatorial interpretation of the Ehrhart h-vector for TKn .

Original languageEnglish
Article number101976
JournalAdvances in Applied Mathematics
StatePublished - Mar 2020

Bibliographical note

Publisher Copyright:
© 2019 Elsevier Inc.

ASJC Scopus subject areas

  • Applied Mathematics


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