The McGee graph is a cubic symmetric graph on 24 nodes and 36 edges which is the unique 7-cage graph.
It can be constructed as the union of the two leftmost subgraphs illustrated above
(Harary 1994, p. 174). It was discovered by H. Sachs (unpublished; see
Kárteszi 1960) and McGee (1960), and proven unique by Tutte (1966; Wong 1982,
Brouwer et al. 1989, p. 209). It has girth 7, diameter 4, and chromatic
number 3.
A symmetric embedding is illustrated above.
The McGee graph is Hamiltonian and has a single distinct LCF notation of order 8, , one of order 2, and two of order 1, all illustrated
above.
The graph is not vertex-transitive (Holton and Sheehan 1993, pp. 207-208) since its automorphism group has orbits of length
8 and 16. It is therefore also not 4-transitive (as incorrectly stated by Harary
1994, p. 175).
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