# Laplacian Simplices

## TL;DR

This paper introduces the concept of Laplacian simplices derived from graphs, exploring their properties such as reflexivity, the integer decomposition property, and unimodality of Ehrhart $h^*$-vectors, with results for various graph classes.

## Contribution

It systematically studies Laplacian simplices for different graphs, establishing new properties and providing explicit Ehrhart $h^*$-vector formulas, especially for cycles and complete graphs.

## Key findings

- Laplacian simplices are reflexive for trees, odd cycles, complete graphs, and whiskered even cycles.
- The $T_{K_n}$ simplices have the integer decomposition property, while $T_{C_n}$ for odd cycles do not.
- Unimodality of Ehrhart $h^*$-vectors is shown for trees, odd cycles, and complete graphs.

## Abstract

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 $T_{K_n}$.