McGee graph

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McGee graph
McGee graph hamiltonian.svg
The McGee graph
Named after W. F. McGee
Vertices 24
Edges 36
Radius 4
Diameter 4[1]
Girth 7[1]
Automorphisms 32[1]
Chromatic number 3[1]
Chromatic index 3[1]
Properties Cubic

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 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First discovered by Sachs but unpublished,[2] the graph is named after McGee who published the result in 1960.[3] Then, the McGee graph was the proven the unique (3,7)-cage by Tutte in 1966.[4][5][6]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 five non-isomorphic graphs tied for being the smallest cubic graph that requires eight crossings. Another of these five 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.

Algebraic properties[edit]

The characteristic polynomial of the McGee 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 graph is the smallest cubic cage that is not a vertex-transitive graph.[9]



  1. ^ a b c d e f 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. "A Minimal Cubic Graph of Girth Seven." Canad. Math. Bull. 3, 149-152, 1960
  4. ^ Tutte, W. T. Connectivity in Graphs. Toronto, Ontario: University of Toronto Press, 1966
  5. ^ Wong, P. K. "Cages--A Survey." J. Graph Th. 6, 1-22, 1982
  6. ^ Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance Regular Graphs. New York: Springer-Verlag, p. 209, 1989
  7. ^ "Sloane's A110507 ", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation..
  8. ^ Pegg, E. T.; Exoo, G. (2009), "Crossing number graphs", Mathematica Journal 11 .
  9. ^ Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.