McGee graph

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.