Hypercube graph

In graph theory, the hypercube graph Q_n is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q₃ is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Q_n has 2ⁿ vertices, 2^{n – 1}n edges, and is a regular graph with n edges touching each vertex.

Hypercube graph
The hypercube graph Q4
Vertices	2n
Edges	2n – 1n
Diameter	n
Girth	4 if n ≥ 2
Automorphisms	n! 2n
Chromatic number	2
Spectrum	${\displaystyle \{(n-2k)^{\binom {n}{k}};k=0,\ldots ,n\}}$
Properties	Symmetric Distance regular Unit distance Hamiltonian Bipartite
Notation	Qn
Table of graphs and parameters

The hypercube graph Q_n may also be constructed by creating a vertex for each subset of an n-element set, with two vertices adjacent when their subsets differ in a single element, or by creating a vertex for each n-digit binary number, with two vertices adjacent when their binary representations differ in a single digit. It is the n-fold Cartesian product of the two-vertex complete graph, and may be decomposed into two copies of Q_{n – 1} connected to each other by a perfect matching.

Hypercube graphs should not be confused with cubic graphs, which are graphs that have exactly three edges touching each vertex. The only hypercube graph Q_n that is a cubic graph is the cubical graph Q₃.