page_title: Vertex-transitive graph
section_title: Finite examples
hierarchical_section_title: Vertex-transitive graph / Finite examples
caption_attribution_description: English: en:Schlegel diagram
context_page_description: In the mathematical field of graph theory, a vertex-transitive graph is a graph G in which, 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, and not all regular graphs are vertex-transitive.
context_section_description: 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.
caption: The edges of the truncated tetrahedron form a vertex-transitive graph (also a Cayley graph) which is not symmetric.
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