An automorphic graph is a distance-transitive graph for which the automorphism group acts primitively on the vertices of and is not a complete graph or an line graph (Biggs 1993, p. 178).
The following table summarizes some known automorphic graphs (Gordon and Levingston 1981; Biggs 1976; Biggs 1993, pp. 178-179). There are exactly three cubic automorphic graphs, and a single quartic automorphic graph (Biggs 1976). Note that odd graphs are automorphic (Biggs 1976).
degree | graph | intersection array | |
10 | 3 | Petersen graph | |
28 | 3 | Coxeter graph | |
35 | 4 | odd graph | |
36 | Sylvester graph | ||
57 | Perkel graph | ||
63 | Conway-Smith graph | ||
65 | Hall graph | ||
68 | Doro graph | ||
102 | 3 | Biggs-Smith graph | |
126 | 5 | odd graph | |
208 | |||
266 | |||
280 | |||
330 | |||
462 | 6 | odd graph | |
5 | |||