Laplacian simplices

This paper initiates the study of the Laplacian simplex T_G 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 T_G 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 T_G is reflexive. We show that while T_Kn has the integer decomposition property, T_Cn for odd cycles does not. The Ehrhart h^⁎-vectors of T_G 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 T_Cn is given by (h₀ ^⁎,…,h_n−1 ^⁎)=(1,…,1,n²−n+1,1,…,1). We also provide a combinatorial interpretation of the Ehrhart h^⁎-vector for T_Kn .