The McGee graph is a cubic graph on 24 vertices and 36 edges which is the unique 7-cage graph. It can be constructed as the graph 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 It was discovered by H. Sachs (unpublished; see Kárteszi 1960) and McGee (1960), and proven unique by Tutte (1966), Wong (1982), and Brouwer et al. (1989, p. 209). It has girth 7, graph diameter 4, and chromatic number 3.
A graph drawing with symmetry is illustrated above.
The McGee graph is Hamiltonian and has a single distinct LCF notation of order 8, ![[-12,7,-7]^8](/images/equations/McGeeGraph/Inline1.svg)
, one of order 2, and two of order 1, all illustrated
above.
Its automorphism group is of group order 32 and it has graph spectrum
 |
where 
, 
, and 
are the polynomial roots
of 
.
The McGee graph has rectilinear crossing number 8, as established by G. Exoo around 1990 (G. Exoo, pers. comm., May 12, 2019). It is one of three cubic graphs on 24 vertices with smallest possible graph crossing number of 8 (another being the Nauru graph), making it a smallest cubic crossing number graph (Pegg and Exoo 2009, Clancy et al. 2020).
The graph is not vertex-transitive (Holton and Sheehan 1993, pp. 207-208) since its graph automorphism group has orbits of length 8 and 16. It is therefore also not 4-transitive (as incorrectly stated by Harary 1994, p. 175).
It is implemented in the Wolfram Language as GraphData["McGeeGraph"].
