The Vertex-Transitive Graphs on 14 Vertices

Last update=23 May, 2006 There are 51 connected vertex-transitive 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:

• 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
• BiDbl(G)   means the bipartite double of G. Make 2 copies of V(G), call them u1,...,un and v1,...,vn. If uv is an edge of G, then u1v2 and v1u2 are edges of BiDbl(G)
• 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 and the complements of the disconnected transitive graphs are:

VT12_39 = ~C14(4)
VT12_40 = ~2C7(2)
VT12_41 = ~C14(2)
VT12_42 = ~C14(6)=~Dbl(C7)
VT12_43 = ~antip(Heawood)
VT12_44 = ~C14(3)=~BiDbl(C7(2))
VT12_45 = ~(C7xK2)=~Prism(7)
VT12_46 = ~C14+
VT12_47 = ~Heawood
VT12_48 = ~2C7
VT12_49 = ~C14
VT12_50 = ~7K2
VT12_51 = ~K7

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 K7 on the torus, which is basically how Heawood discovered it.          Back to the Groups & Graphs home page.