# McGee graph

McGee Graph
The McGee Graph
Named after W. F. McGee
Vertices 24
Edges 36
Diameter 4[1]
Girth 7[1]
Automorphisms 32[1]
Chromatic number 3[1]
Chromatic index 3[1]
Properties Cubic
Cage
Hamiltonian

In the mathematical field of graph theory, the McGee Graph or the (3-7)-cage is a 3-regular graph with 24 vertices and 36 edges.[1]

The McGeeGraph is the unique (3,7)-cage (the smallest cubic graph of girth 7). It is also the smallest cubic cage that is not a Moore graph.

First discovered by Sachs but unpublished,[2] the graph is named after McGee who published the result in 1960.[3] Then, the McGee graph was the proven the unique (3,7)-cage by Tutte in 1966.[4][5][6]

The smallest cubic graphs with crossing numbers 1–8 are known (sequence A110507 in OEIS). One of the smallest 8-crossing graphs is the McGee graph. There exists 5 non-isomorphic cubic graphs of order 24 with crossing number 8.[7] One of them is the generalized Petersen graph G(12,5), also known as the Nauru graph.[8]

The McGeeGraph has radius 4, diameter 4, chromatic number 3 and chromatic index 3. It is also a 3-vertex-connected and a 3-edge-connected graph.

## Algebraic properties

The characteristic polynomial of the McGeeGraph graph is : $x^3(x-3)(x-2)^3(x+1)^2(x+2)(x^2+x-4)(x^3+x^2-4x-2)^4$.

The automorphism group of the McGee graph is of order 32 and doesn't acts transitively upon its vertices: there are two vertex orbits of lengths 8 and 16. The McGee is the smallest cubic cage that is not a vertex-transitive graph.[9]

## References

1. ^ Kárteszi, F. "Piani finit ciclici come risoluzioni di un certo problemo di minimo." Boll. Un. Mat. Ital. 15, 522-528, 1960
2. ^ McGee, W. F. "A Minimal Cubic Graph of Girth Seven." Canad. Math. Bull. 3, 149-152, 1960
3. ^ Tutte, W. T. Connectivity in Graphs. Toronto, Ontario: University of Toronto Press, 1966
4. ^ Wong, P. K. "Cages--A Survey." J. Graph Th. 6, 1-22, 1982
5. ^ Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance Regular Graphs. New York: Springer-Verlag, p. 209, 1989
6. ^ Pegg, E. T. and Exoo, G. "Crossing Number Graphs." Mathematica J. 11, 2009
7. ^
8. ^ Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.