The VertexTransitive Graphs on 14 Vertices
Last update=23 May, 2006
There are 51 connected vertextransitive graphs on 14 vertices. There are 3 of degree 3 (21 edges), and 5 of degree 4 (28 edges). These 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
 BiDbl(G)^{ }_{ } means the bipartite double of G. Make 2 copies of V(G), call them u_{1},...,u_{n} and v_{1},...,v_{n}. If uv is an edge of G, then u_{1}v_{2} and v_{1}u_{2} are edges of BiDbl(G)
 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 and the complements of the disconnected transitive graphs are:
 VT12_39 = ~C_{14}(4)
 VT12_40 = ~2C_{7}(2)
 VT12_41 = ~C_{14}(2)
 VT12_42 = ~C_{14}(6)=~Dbl(C_{7})
 VT12_43 = ~antip(Heawood)
 VT12_44 = ~C_{14}(3)=~BiDbl(C_{7}(2))
 VT12_45 = ~(C_{7}xK_{2})=~Prism(7)
 VT12_46 = ~C_{14}^{+}
 VT12_47 = ~Heawood
 VT12_48 = ~2C_{7}
 VT12_49 = ~C_{14}
 VT12_50 = ~7K_{2}
 VT12_51 = ~K_{7}
The Heawood graph is the incidence graph of the Fano plane, the unique projective plane with 7 points and 7 lines. The Heawood graph is also known as the (3,6)cage. It is also the dual of the unique embedding of K_{7} on the torus, which is basically how Heawood discovered it.
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