Two vertices (or edges) of a periodic graph are symmetric if they are in the same orbit of the symmetry group of the graph; in other words, two vertices (or edges) are symmetric if there is a symmetry of the net that moves one onto the other. In chemistry, there is a tendency to refer to orbits of vertices or edges as “kinds” of vertices or edges, with the recognition that from any two vertices or any two edges (similarly oriented) of the same orbit, the geometric graph “looks the same”. Finite colorings of vertices and edges (where symmetries are to preserve colorings) may be employed.
The symmetry group of a crystal net will be a (group of restrictions of a) crystallographic space group, and many of the most common crystals are of very high symmetry, i.e. very few orbits. A crystal net is uninodal if it has one orbit of vertex (if the vertices were colored and the symmetries preserve colorings, this would require that a corresponding crystal have atoms of one element or molecular building blocks of one compound – but not vice versa, for it is possible to have a crystal of one element but with several orbits of vertices). Crystals with uninodal crystal nets include cubic diamond and some representations of quartz crystals. Uninodality corresponds with isogonality in geometry and vertex-transitivity in graph theory, and produces examples objective structures. A crystal net is binodal if it has two orbits of vertex; crystals with binodal crystal nets include boracite and anatase. It is edge-transitive or isotoxal if it has one orbit of edges; crystals with edge-transitive crystal nets include boracite but not anatase – which has two orbits of edges.