  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 $T_{K_n}$ has the integer decomposition property, $T_{C_n}$ 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 $T_{C_n}$ is given by
$(h_0^*,\ldots,h_{n-1}^*)=(1,\ldots,1,n^2-n+1,1,\ldots, 1)$. We also provide a
combinatorial interpretation of the Ehrhart $h^*$-vector for $T_{K_n}$.
