McGee graph

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]

McGee graph
The McGee graph
Named afterW. F. McGee
Vertices24
Edges36
Radius4
Diameter4[1]
Girth7[1]
Automorphisms32[1]
Chromatic number3[1]
Chromatic index3[1]
Book thickness3
Queue number2
PropertiesCubic
Cage
Hamiltonian
Table of graphs and parameters

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First discovered by Sachs but unpublished,[2] the graph is named after McGee who published the result in 1960.[3] Then, the McGee graph was proven the unique (3,7)-cage by Tutte in 1966.[4][5][6]

The McGee graph requires at least eight crossings in any drawing of it in the plane. It is one of three non-isomorphic graphs tied for being the smallest cubic graph that requires eight crossings. Another of these three graphs is the generalized Petersen graph G(12,5), also known as the Nauru graph.[7][8]

The McGee graph 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. It has book thickness 3 and queue number 2.[9]

Algebraic properties

The characteristic polynomial of the McGee graph is

.

The automorphism group of the McGee graph is of order 32 and doesn't act transitively upon its vertices: there are two vertex orbits, of lengths 8 and 16. The McGee graph is the smallest cubic cage that is not a vertex-transitive graph.[10]

References
  1. Weisstein, Eric W. "McGee Graph". MathWorld.
  2. Kárteszi, F. "Piani finit ciclici come risoluzioni di un certo problemo di minimo." Boll. Un. Mat. Ital. 15, 522-528, 1960
  3. McGee, W. F. (1960), "A Minimal Cubic Graph of Girth Seven", Canadian Mathematical Bulletin, 3 (2): 149–152, doi:10.4153/CMB-1960-018-1
  4. Tutte, W. T. Connectivity in Graphs. Toronto, Ontario: University of Toronto Press, 1966
  5. Wong, Pak-Ken (1982), "Cages—A Survey", Journal of Graph Theory, 6: 1–22, doi:10.1002/jgt.3190060103
  6. Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance Regular Graphs. New York: Springer-Verlag, p. 209, 1989
  7. Sloane, N. J. A. (ed.). "Sequence A110507 (Number of nodes in the smallest cubic graph with crossing number n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  8. Pegg, E. T.; Exoo, G. (2009), "Crossing number graphs", Mathematica Journal, 11 (2), doi:10.3888/tmj.11.2-2.
  9. Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
  10. Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.
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