Symmetric graph

of G, there is an automorphism such that In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction).

[3] By definition (ignoring u1 and u2), a symmetric graph without isolated vertices must also be vertex-transitive.

[1] Since the definition above maps one edge to another, a symmetric graph must also be edge-transitive.

Star graphs are a simple example of being edge-transitive without being vertex-transitive or symmetric.

As a further example, semi-symmetric graphs are edge-transitive and regular, but not vertex-transitive.

[3] However, for even degree, there exist connected graphs which are vertex-transitive and edge-transitive, but not symmetric.

Further symmetric graphs are formed by the vertices and edges of the regular and quasiregular polyhedra: the cube, octahedron, icosahedron, dodecahedron, cuboctahedron, and icosidodecahedron.

Extension of the cube to n dimensions gives the hypercube graphs (with 2n vertices and degree n).

Combining the symmetry condition with the restriction that graphs be cubic (i.e. all vertices have degree 3) yields quite a strong condition, and such graphs are rare enough to be listed.

The Foster census and its extensions provide such lists.

In the case of the degree being exactly 3 (cubic symmetric graphs), there are none for t ≥ 6.