  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 and
Packing Radius. We use this information to obtain the following results: i.
Every multigraph defines a Delaunay triangulation of its Laplacian lattice and
this Delaunay triangulation contains complete information of the multigraph up
to isomorphism. ii. 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 formulas 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 good covering and packing
properties.
