In geometry and crystallography, the Laves graph is an infinite and highly symmetric system of points and line segments in three-dimensional Euclidean space, forming a periodic graph. Three equal-length segments meet at 120° angles at each point, and all cycles use ten or more segments. It is the shortest possible triply periodic graph, relative to the volume of its fundamental domain. One arrangement of the Laves graph uses one out of every eight of the points in the integer lattice as its points, and connects all pairs of these points that are nearest neighbors, at distance ${\displaystyle {\sqrt {2}}}$. It can also be defined, divorced from its geometry, as an abstract undirected graph, a covering graph of the complete graph on four vertices.

H. S. M. Coxeter (1955) named this graph after Fritz Laves, who first wrote about it as a crystal structure in 1932. It has also been called the K₄ crystal, (10,3)-a network, diamond twin, triamond, and the srs net. The regions of space nearest each vertex of the graph are congruent 17-sided polyhedra that tile space. Its edges lie on diagonals of the regular skew polyhedron, a surface with six squares meeting at each integer point of space.

Several crystalline chemicals have known or predicted structures in the form of the Laves graph. Thickening the edges of the Laves graph to cylinders produces a related minimal surface, the gyroid, which appears physically in certain soap film structures and in the wings of butterflies.

As Coxeter (1955) describes, the vertices of the Laves graph can be defined by selecting one out of every eight points in the three-dimensional integer lattice, and forming their nearest neighbor graph. Specifically, one chooses the points $${\displaystyle {\begin{aligned}(0,0,0),\quad (1,2,3),\quad (2,3,1),\quad (3,1,2),\\(2,2,2),\quad (3,0,1),\quad (0,1,3),\quad (1,3,0),\\\end{aligned}}}$$ and all the other points formed by adding multiples of four to these coordinates. The edges of the Laves graph connect pairs of points whose Euclidean distance from each other is the square root of two, ${\displaystyle {\sqrt {2}}}$, as the points of each pair differ by one unit in two coordinates, and are the same in the third coordinate. The edges meet at 120° angles at each vertex, in a flat plane. All pairs of vertices that are non-adjacent are farther apart, at a distance of at least ${\displaystyle {\sqrt {6}}}$ from each other. The edges of the resulting geometric graph are diagonals of a subset of the faces of the regular skew polyhedron with six square faces per vertex, so the Laves graph is embedded in this skew polyhedron.

It is possible to choose a larger set of one out of every four points of the integer lattice, so that the graph of distance-${\displaystyle {\sqrt {2}}}$ pairs of this larger set forms two mirror-image copies of the Laves graph, disconnected from each other, with all other pairs of points farther than ${\displaystyle {\sqrt {2}}}$ apart.

As an abstract graph, the Laves graph can be constructed as the maximal abelian covering graph of the complete graph ${\displaystyle K_{4}}$. Being an abelian covering graph of ${\displaystyle K_{4}}$ 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 ${\displaystyle K_{4}}$ involving a higher-dimensional lattice. This construction justifies an alternative name of the Laves graph, the ${\displaystyle K_{4}}$ crystal.

A maximal abelian covering graph can be constructed from any finite graph ${\displaystyle G}$; applied to ${\displaystyle K_{4}}$, the construction produces the (abstract) Laves graph, but does not give it the same geometric layout. Choose a spanning tree of ${\displaystyle G}$, let ${\displaystyle d}$ 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 ${\displaystyle \mathbb {Z} ^{d}}$ for each of these non-tree edges. Then, fix the set of vertices of the covering graph to be the ordered pairs ${\displaystyle (v,w)}$ where ${\displaystyle v}$ is a vertex of ${\displaystyle G}$ and ${\displaystyle w}$ is a vector in ${\displaystyle \mathbb {Z} ^{d}}$. For each such pair, and each edge ${\displaystyle uv}$ adjacent to ${\displaystyle v}$ in ${\displaystyle G}$, make an edge from ${\displaystyle (v,w)}$ to ${\displaystyle (u,w\pm \epsilon )}$ where ${\displaystyle \epsilon }$ is the zero vector if ${\displaystyle uv}$ belongs to the spanning tree, and is otherwise the basis vector associated with ${\displaystyle uv}$, 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 ${\displaystyle d}$-dimensional integer lattice (as a graph with unit-length edges) is the maximal abelian covering graph of a graph with one vertex and ${\displaystyle d}$ self-loops.