The Laplacian lattice of a graph under a simplicial distance function

TL;DR

This paper characterizes geometric invariants of Laplacian lattices of multigraphs under a simplicial distance, revealing how these invariants encode graph structure and influence lattice properties.

Contribution

It provides a complete geometric description of Laplacian lattices, linking Delaunay triangulations to graph isomorphism and deriving bounds on multigraph counts and lattice densities.

Findings

Delaunay triangulation encodes multigraph isomorphism.

Number of multigraphs bounded by Delaunay triangulations.

Highly connected multigraphs have optimal covering and packing densities.

Abstract

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.