Laplacian simplices

Abstract

This paper initiates the study of the Laplacian simplexTG 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.