The Laplacian lattice of a graph under a simplicial distance function

We provide a complete description of important geometric invariants of the Laplacian lattice of a multigraph under the distance function induced by a regular simplex, namely Voronoi Diagram, Delaunay Triangulation, Delaunay Polytope and its combinatorial structure, Shortest Vectors, Covering Radius and Packing Radius. We use this information to obtain the following results: i. We provide an explicit description of the vertices, edges and facets of the Delaunay polytope of the origin under the simplicial distance function and using the description, we show that the Delaunay triangulation contains complete information of the multigraph up to isomorphism. ii. A weak generalization of the matrix-tree theorem: the number of multigraphs with a given Laplacian lattice is controlled, in particular upper bounded, by the number of different Delaunay triangulations. iii. We obtain formulae for the covering and packing densities of a Laplacian lattice and deduce that in the space of Laplacian lattices of undirected connected multigraphs, the Laplacian lattices of highly connected multigraphs such as Ramanujan multigraphs possess high covering and packing properties in the space of Laplacian lattices of connected graphs.