Cube graph

This article defines a particular undirected graph, i.e., the definition here determines the graph uniquely up to graph isomorphism.
View a complete list of particular undirected graphs

Definition

The term cube graph, sometimes cube or graph of a cube, can be defined in the following equivalent ways:

1. It is the cube in three-dimensional space viewed as a graph: the vertices are the vertices of the cube, and the edges are the edges of the cube. Alternatively, it can be thought of as the $n$-dimensional hypercube graph where $n=3$.
2. It is the prism of cycle graph:C4, or equivalently, it is the dihedral graph on 8 vertices.

Terminological confusion

This is not to be confused with cubic graph, a term used for a 3-regular graph. Note that the cube graph is cubic, but is not the only cubic graph.

Arithmetic functions

Size measures

Function	Value	Explanation
size of vertex set	8	As $n$-dimensional hypercube, $n=3$: $2^{n}=2^{3}=8$
size of edge set	12	As $n$-dimensional hypercube, $n=3$: $n\cdot 2^{n-1}=3\cdot 2^{3-1}=12$

Numerical invariants associated with vertices

Since the graph is a vertex-transitive graph, any numerical invariant associated to a vertex must be equal on all vertices of the graph. Below are listed some of these invariants:

Function	Value	Explanation
degree of a vertex	3	As $n$-dimensional hypercube, $n=3$: $n=3$
eccentricity of a vertex	3	As $n$-dimensional hypercube, $n=3$: $n=3$

Other numerical invariants

Function	Value	Explanation
clique number	2	As $n$-dimensional hypercube, $n=3$: 2 (independent of $n$)
independence number	4	As $n$-dimensional hypercube, $n=3$: $2^{n-1}=2^{3-1}=4$
chromatic number	2	As $n$-dimensional hypercube, $n=3$: 2 (independent of $n$
radius of a graph	3	Due to vertex-transitivity, the radius equals the eccentricity of any vertex, which has been computed above.
diameter of a graph	3	Due to vertex-transitivity, the radius equals the eccentricity of any vertex, which has been computed above.
odd girth	infinite
even girth	4	As $n$-dimensional hypercube, $n=3$: 4 (independent of $n$ for $n\geq 2$)
girth of a graph	4	As $n$-dimensional hypercube, $n=3$: 4 (independent of $n$ for $n\geq 2$)

Graph properties

Property	Satisfied?	Explanation
connected graph	Yes
regular graph	Yes	all vertices have degree three
vertex-transitive graph	Yes
cubic graph	Yes	all vertices have degree three
edge-transitive graph	Yes
symmetric graph	Yes
distance-transitive graph	Yes
bridgeless graph	Yes
strongly regular graph	No
bipartite graph	Yes