Graph families defined by their automorphisms | ||||
---|---|---|---|---|
distance-transitive | → | distance-regular | ← | strongly regular |
↓ | ||||
symmetric (arc-transitive) | ← | t-transitive, t ≥ 2 | skew-symmetric | |
↓ | ||||
(if connected) vertex- and edge-transitive |
→ | edge-transitive and regular | → | edge-transitive |
↓ | ↓ | ↓ | ||
vertex-transitive | → | regular | → |
(if bipartite) biregular |
↑ | ||||
Cayley graph | ← | zero-symmetric | asymmetric |
In the mathematical field of graph theory, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism
such that
In other words, a graph is vertex-transitive if its automorphism group acts transitively upon its vertices. A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical.
Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular. However, not all vertex-transitive graphs are symmetric (for example, the edges of the truncated tetrahedron), and not all regular graphs are vertex-transitive (for example, the Frucht graph and Tietze's graph).
Finite vertex-transitive graphs include the symmetric graphs (such as the Petersen graph, the Heawood graph and the vertices and edges of the Platonic solids). The finite Cayley graphs (such as cube-connected cycles) are also vertex-transitive, as are the vertices and edges of the Archimedean solids (though only two of these are symmetric). Potočnik, Spiga and Verret have constructed a census of all connected cubic vertex-transitive graphs on at most 1280 vertices.
Although every Cayley graph is vertex-transitive, there exist other vertex-transitive graphs that are not Cayley graphs. The most famous example is the Petersen graph, but others can be constructed including the line graphs of edge-transitive non-bipartite graphs with odd vertex degrees.