The VertexTransitive Graphs on 16 Vertices
Last update=6 Feb, 2009
There are 272 connected vertextransitive graphs on 16 vertices. A selection of them is 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
 L(G)^{ }_{ } means the line graph of G
 Q_{3} and Q_{4} ^{ }_{ } mean the graphs of the cube and hypercube
 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}
 BiDbl(G)^{ }_{ } means the bipartite double of G. Make 2 copies of V(G), call them V_{1} and V_{2}. If uv is an edge of G, then u_{1}v_{2} and v_{1}u_{2} are edges of BiDbl(G)
 BiDbl^{+}(G)^{ }_{ } means the bipartite double of G, with the additional edges u_{1}u_{2} for each u in V(G)
 SR(n,k,λ,μ) means a strongly regular graph with n vertices, degree k, and parameters λ and μ
A number of the transitive graphs on 16 vertices can be constructed from 2Q_{3}, two copies of the cube. Then add additional edges. For example Q_{4}, the Clebsch graph, and the Shrikande graph can all be constructed
in this fashion. It is possible to chose the vertices of 2Q_{3} in four
groups of four, and induce K_{4}'s on these subsets, etc., to obtain a
number of vertextransitive graphs.
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