The Vertex-Transitive Graphs on 12 Vertices

Last update=20 May, 2006


There are 64 connected vertex-transitive graphs on 12 vertices. The 10 of degree 4 (hence 24 edges) are shown here. The order of the automorphism group is given in square brackets in each window's title.


  • Cn means the cycle of length n
  • Cn+ means the cycle of length n with diagonals
  • Cn(k)  means the cycle of length n with chords of length k
  • Cn(k+)  means the cycle of length n with chords of length k from every second vertex
  • ~G   means the complement of G
  • 2G   means two disjoint copies of G
  • GxH   means the direct product of G and H
  • Prism(m)  means CmxK2, ie, two cycles with corresponding vertices joined by a matching
  • trunc(G),  where G is planar, means to truncate G, ie, replace each vertex of degree k by Ck
  • L(G)   means the line-graph of G
  • Octahedron   means the graph of the octahedron; this is L(K4) or ~3K2 or C6(2)
  • Dbl(G)   means the double of G. Make 2 copies of G, call them G1 and G2. If uv is an edge of G, then u1v2 and v1u2 are also edges of Dbl(G)
  • Dbl+(G)   means the double of G, with the additional edges u1u2
  • antip(G)  means the antipodal graph of G. It has the same vertices, but u and v are joined in antip(G) if they are are maximum distance in G

The complements of the graphs shown here are:

VT12_41 = ~C12(5+,6)
VT12_42 = ~C12(4)
VT12_43 = ~C12(2)
VT12_44 = ~L(Cube)
VT12_45 = ~antip(trunc(K4))
VT12_46 = ~2Octahedron
VT12_47 = ~(C4xC3)
VT12_48 = ~C12(3+,6)
VT12_49 = ~C12(3)
VT12_50 = ~(K3,3xK2)
VT12_51 = ~C12(5)=Dbl+(Prism(3))

VT12_6 VT12_7
VT12_8 VT12_9
VT12_10 VT12_11
VT12_12 VT12_13
VT12_14 VT12_15


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