McGee graph
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]
Book thickness	3
Queue number	2
Properties	Cubic Cage Hamiltonian
Table of graphs and parameters

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First discovered by Sachs but unpublished, ^[2] the graph is named after McGee who published the result in 1960.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 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]

The characteristic polynomial of the McGee graph is

${\displaystyle x^{3}(x-3)(x-2)^{3}(x+1)^{2}(x+2)(x^{2}+x-4)(x^{3}+x^{2}-4x-2)^{4}}$.

The automorphism group of the McGee graph is of order 32 and doesn't 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]

• 
  The crossing number of the McGee graph is 8.
• 
  The chromatic number of the McGee graph is 3.
• 
  The chromatic index of the McGee graph is 3.
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  The acyclic chromatic number of the McGee graph is 3.
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  Alternative drawing of the McGee graph.