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.
A substantial body of work is devoted to the study of lattices constructed from graphs, the most well studied ones being the lattice of integral cuts and the lattice of integral flows. This line of investigation was pioneered by the work of Bacher, Harpe and Nagnibeda @cite_3 where they provide a combinatorial interpretation of various parameters of the lattice of integral flows and the lattice of integral cuts in the Euclidean distance function, for example they show that the square norm of the shortest vector of the lattice of integral flows is equal to the girth of the graph. In contrast, the study of the Laplacian lattice under simplicial distance function gives rise to some new and interesting phenomenon, that are not seen in the Euclidean case and in some cases provides more refined information about the graph. Here we note such instances:
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Les flots entiers sur un graphe fini Γ constituent naturellement un reseau entier Λ 1 (Γ) dans l'espace euclidien Ker(Δ 1 ) des fonctions harmoniques a valeurs reelles sur l'ensemble des aretes de Γ. On montre l'equivalence de diverses proprietes de Γ (caractere biparti, tour de taille, complexite, separabilite) avec des proprietes convenables de Λ l (Γ) (parite, norme minimales determinant, decomposabilite). Le reseau dual de Λ 1 (Γ) est identifie a la cohomologie enliere H 1 (Γ,Z), plongee dans Ker(Δ 1 ). On montre des traductions analogues pour le reseau des coupures entieres et les proprietes convenables du graphe (caractere eulerien, connectivite d'aretes, complexite, separabilite). Ces reseaux ont un groupe determinant qui joue pour les graphes le meme role que la jacobienne pour une surface de Riemann close. Ce sont alors les fonctions harmoniques sur un graphe (a valeurs dans un groupe abelien) qui tiennent lieu d'applications holomorphes.