text: The Laves graph is a cubic graph, meaning that there are exactly three edges at each vertex. Every pair of a vertex and adjacent edge can be transformed into every other such pair by a symmetry of the graph, so it is a symmetric graph. More strongly, for every two vertices  and , every one-to-one correspondence between the three edges incident to  and the three edges incident to  can be realized by a symmetry. However, the overall structure is chiral: no sequence of translations and rotations can make it coincide with its mirror image. The symmetry group of the Laves graph is the space group .
  The girth of this structure is 10—the shortest cycles in the graph have 10 vertices—and 15 of these cycles pass through each vertex. The numbers of vertices at distance 0, 1, 2, ... from any vertex (forming the coordination sequence of the Laves graph) are:
  If the surrounding space is partitioned into the regions nearest each vertex—the cells of the Voronoi diagram of this structure—these form heptadecahedra with 17 faces each. They are plesiohedra, polyhedra that tile space isohedrally. Experimenting with the structures formed by these polyhedra led physicist Alan Schoen to discover the gyroid minimal surface, which is topologically equivalent to the surface obtained by thickening the edges of the Laves graph to cylinders and taking the boundary of their union.
  The Laves graph is the unique shortest triply-periodic network, in the following sense. Triply-periodic means repeating infinitely in all three dimensions of space, so a triply-periodic network is a connected geometric graph with a three-dimensional lattice of translational symmetries. A fundamental domain is any shape that can tile space with its translated copies under these symmetries. Any lattice has infinitely many choices of fundamental domain, of varying shapes, but they all have the same volume . One can also measure the length of the edges of the network within a single copy of the fundamental domain; call this number . Similarly to ,  does not depend on the choice of fundamental domain, as long as the domain boundary only crosses the edges, rather than containing parts of their length. The Laves graph has four symmetry classes of vertices (orbits), because the symmetries considered here are only translations, not the rotations needed to map these four classes into each other. Each symmetry class has one vertex in any fundamental domain, so the fundamental domain contains twelve half-edges, with total length . The volume of its fundamental domain is 32. From these two numbers, the ratio  (a dimensionless quantity) is therefore . This is in fact the minimum possible value: All triply-periodic networks have  with equality only in the case of the Laves graph.