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 .
|Journal||Advances in Applied Mathematics|
|State||Published - Mar 2020|
Bibliographical noteFunding Information:
The first author was partially supported by grant H98230-16-1-0045 from the U.S. National Security Agency. The authors thank Tefjol Pllaha, Liam Solus, Akiyoshi Tsuchiya, and Devin Willmott for their comments and suggestions.
© 2019 Elsevier Inc.
ASJC Scopus subject areas
- Applied Mathematics