McGee graph

McGee graph
The McGee graph
Named after	W. F. McGee
Vertices	24
Edges	36
Radius	4
Diameter	41
Girth	71
Automorphisms	321
Chromatic number	31
Chromatic index	31
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First discovered by Sachs but unpublished,² the graph is named after McGee who published the result in 1960.³ Then, the McGee graph was proven the unique (3,7)-cage by Tutte in 1966.⁴⁵⁶

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.⁷⁸

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 graph is 1-planar.¹⁰

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.¹¹

The automorphism group of the McGee graph, meaning its group of symmetries, has 32 elements. This group is isomorphic to the group of all affine transformations of ${\displaystyle \mathbb {Z} /8\mathbb {Z} }$, i.e., transformations of the form

${\displaystyle x\mapsto ax+b}$

where ${\displaystyle a,b\in \mathbb {Z} /8\mathbb {Z} }$ and ${\displaystyle a}$ is invertible, so ${\displaystyle a=1,3,5,7}$.¹² This is one of the two smallest possible group ${\displaystyle G}$ with an outer automorphism that maps every element ${\displaystyle g\in G}$ to an element conjugate to ${\displaystyle g}$.¹³

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  The crossing number of the McGee graph is 8.
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  The chromatic number of the McGee graph is 3.
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  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.