The VertexTransitive Graphs on 12 Vertices
Last update=20 May, 2006
There are 64 connected vertextransitive 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.
Notation:
 C_{n} means the cycle of length n
 C_{n}^{+} means the cycle of length n with diagonals
 C_{n}(k)^{ } means the cycle of length n with chords of length k
 C_{n}(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 C_{m}xK_{2}, 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 C_{k}
 L(G)^{ }_{ } means the linegraph of G
 Octahedron^{ }_{ } means the graph of the octahedron; this is L(K_{4}) or ~3K_{2} or C_{6}(2)
 Icosahedron^{ }_{ } means the graph of the icosahedron
 Dbl(G)^{ }_{ } means the double of G. Make 2 copies of G, call them G_{1} and G_{2}. If uv is an edge of G, then u_{1}v_{2} and v_{1}u_{2} are also edges of Dbl(G)
 Dbl^{+}(G)^{ }_{ } means the double of G, with the additional edges u_{1}u_{2}
 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 = ~2K_{6}=Dbl(K_{3,3})
 VT12_29 = ~(K_{3}xK_{4})
 VT12_30 = ~Icosahedron
 VT12_31 = C_{12}(3,4)=~C_{12}(2,6)
 VT12_32 = ~C_{12}(5,6)=Dbl(Prism(3))
 VT12_33 = C_{12}(2,4)=~C_{12}(3,6)
 VT12_34 = L(Octahedron)=L(L(K_{4}))
 VT12_35 = ~C_{12}(2,5^{+})
 VT12_36 = C_{12}(4,5)
 VT12_37 = ~C_{12}(4,5^{+})
 VT12_38 = ~(OctahedronxK_{2})
 VT12_39 = C_{12}(2,3)=~C_{12}(4,6)
 VT12_40 = K_{6}xK_{2}=L(K_{2,6})
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