McGee graph

Template:Short description Template:Infobox 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.

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First discovered by Sachs but unpublished, the graph is named after McGee who published the result in 1960.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

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 Template:Nobr, also known as the Nauru graph.

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. The graph is 1-planar.

Algebraic properties

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 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.

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 $\mathbb{Z}/8\mathbb{Z}$, i.e., transformations of the form

$x\mapsto ax+b$

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

Gallery

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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.