Fig. 1. The dodecahedron given in Frucht’s notation relative to a (4, 5)-semiregular automorphism.

Fig. 2. A list of all connected non-Cayley vertex-transitive graphs of order 20 that are of valency less then 7 given in Frucht’s notation.

Fig. 3. The vertex-transitive graph whose automorphism group is isomorphic to given in Frucht’s notation relative to a (4, 7)-semiregular automorphism.

Fig. 4. Possible forms of the bipartite graph X[B,B^′] where B and B^′ are adjacent blocks of size 4.

Fig. 5. Two possibilities for a spanning subgraph in X. The graph on the left where corresponds to the graph in Fig. 4(ii) and the graph on the right corresponds to the graph in Fig. 4(iii).

Fig. 6. The six possibilities for the quotient graph X_γ of a connected vertex-transitive graph X of order 4p.

Fig. 7. Frucht’s notation of a graph with symbol .

Fig. 8. The vertex-transitive graph of valency 9 on 28 vertices, with a primitive automorphism group arising from the action of the group on the cosets of a subgroup D₁₈.

Fig. 9. The vertex-transitive graph of valency 6 on 52 vertices with a primitive automorphism group arising from the action of the group acting on the 52=4p incident point–line pairs of PG(2,3).

Eight classes of vertex-transitive graphs of order 4p

Symbols of connected vertex-transitive graphs of valency less than one third of the number of vertices arising from the actions in Proposition 3.1