text: As an abstract graph, the Laves graph can be constructed as the maximal abelian covering graph of the complete graph . Being an abelian covering graph of  means that the vertices of the Laves graph can be four-colored such that each vertex has neighbors of the other three colors and so that there are color-preserving symmetries taking any vertex to any other vertex with the same color. For the Laves graph in its geometric form with integer coordinates, these symmetries are translations that add even numbers to each coordinate (additionally, the offsets of all three coordinates must be congruent modulo four). When applying two such translations in succession, the net translation is irrespective of their order: they commute with each other, forming an abelian group. The translation vectors of this group form a three-dimensional lattice. Finally, being a maximal abelian covering graph means that there is no other covering graph of  involving a higher-dimensional lattice. This construction justifies an alternative name of the Laves graph, the  crystal.
  A maximal abelian covering graph can be constructed from any finite graph ; applied to , the construction produces the (abstract) Laves graph, but does not give it the same geometric layout. Choose a spanning tree of , let  be the number of edges that are not in the spanning tree (in this case, three non-tree edges), and choose a distinct unit vector in  for each of these non-tree edges. Then, fix the set of vertices of the covering graph to be the ordered pairs  where  is a vertex of  and  is a vector in . For each such pair, and each edge  adjacent to  in , make an edge from  to  where  is the zero vector if  belongs to the spanning tree, and is otherwise the basis vector associated with , and where the plus or minus sign is chosen according to the direction the edge is traversed. The resulting graph is independent of the chosen spanning tree, and the same construction can also be interpreted more abstractly using homology.
  Using the same construction, the hexagonal tiling of the plane is the maximal abelian covering graph of the three-edge dipole graph, and the diamond cubic is the maximal abelian covering graph of the four-edge dipole. The -dimensional integer lattice (as a graph with unit-length edges) is the maximal abelian covering graph of a graph with one vertex and  self-loops.