In the mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 field of graph theory

Graph theory

In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

, the McGee Graph or the (3-7)-cage is a 3-regular graph

Regular graph

In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each vertex are equal to each other...

 with 24 vertices and 36 edges.

The McGeeGraph is the unique (3,7)-cage (the smallest cubic graph

Cubic graph

In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs....

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First discovered by Sachs but unpublished, the graph is named after McGee who published the result in 1960.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

The smallest cubic graphs with crossing numbers 1–8 are known . The smallest 8-crossing graph is the McGee graph. There exists 5 non-isomorphic cubic graphs of order 24 with crossing number 8. One of them is the generalized Petersen graph

Generalized Petersen graph

In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. The generalized...

 , also known as the Nauru graph

Nauru graph

In the mathematical field of graph theory, the Nauru graph is a symmetric bipartite cubic graph with 24 vertices and 36 edges. It was named by David Eppstein after the twelve-pointed star in the flag of Nauru....

.

The McGeeGraph has radius 4, diameter 4, chromatic number 3 and chromatic index 3. It is also a 3-vertex-connected

K-vertex-connected graph

In graph theory, a graph G with vertex set V is said to be k-vertex-connected if the graph remains connected when you delete fewer than k vertices from the graph...

 and a 3-edge-connected

K-edge-connected graph

In graph theory, a graph is k-edge-connected if it remains connected whenever fewer than k edges are removed.-Formal definition:Let G =  be an arbitrary graph....

 graph.

Algebraic properties

The characteristic polynomial

Characteristic polynomial

In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....

 of the McGeeGraph graph is : .

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 is the smallest cubic cage that is not a vertex-transitive graph

Vertex-transitive graph

In the mathematical field of graph theory, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphismf:V \rightarrow V\ such thatf = v_2.\...

.