The Vertex-Transitive Graphs on 12 Vertices

Last update=20 May, 2006


There are 64 connected vertex-transitive graphs on 12 vertices. The 12 of degree 5 (hence 30 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)
  • Icosahedron   means the graph of the icosahedron
  • 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_28 = ~2K6=Dbl(K3,3)
VT12_29 = ~(K3xK4)
VT12_30 = ~Icosahedron
VT12_31 = C12(3,4)=~C12(2,6)
VT12_32 = ~C12(5,6)=Dbl(Prism(3))
VT12_33 = C12(2,4)=~C12(3,6)
VT12_34 = L(Octahedron)=L(L(K4))
VT12_35 = ~C12(2,5+)
VT12_36 = C12(4,5)
VT12_37 = ~C12(4,5+)
VT12_38 = ~(OctahedronxK2)
VT12_39 = C12(2,3)=~C12(4,6)
VT12_40 = K6xK2=L(K2,6)

VT12_16 VT12_17
VT12_18 VT12_19
VT12_20 VT12_21
VT12_22 VT12_23
VT12_24 VT12_25
VT12_26 VT12_27


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