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.

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_mxK₂, 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 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(K_3,3)

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(K_2,6)

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