Cube graph: Difference between revisions
(Created page with "{{particular undirected graph}} ==Definition== The term '''cube graph''', sometimes '''cube''' or '''graph of a cube''', refers to the cube in three-dimensional space viewed...") |
No edit summary |
||
| Line 4: | Line 4: | ||
The term '''cube graph''', sometimes '''cube''' or '''graph of a cube''', refers to 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. | The term '''cube graph''', sometimes '''cube''' or '''graph of a cube''', refers to 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. | ||
==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== | ==Arithmetic functions== | ||
| Line 50: | Line 54: | ||
|- | |- | ||
| {{arithmetic function value|girth of a graph|4}} || | | {{arithmetic function value|girth of a graph|4}} || | ||
|} | |||
==Graph properties== | |||
{| class="sortable" border="1" | |||
! Property !! Satisfied? !! Explanation | |||
|- | |||
| [[satisfies property::connected graph]] || Yes || | |||
|- | |||
| [[satisfies property::regular graph]] || Yes || all vertices have degree three | |||
|- | |||
| [[satisfies property::vertex-transitive graph]] || Yes || | |||
|- | |||
| [[satisfies property::cubic graph]] || Yes || all vertices have degree three | |||
|- | |||
| [[satisfies property::edge-transitive graph]] || Yes || | |||
|- | |||
| [[satisfies property::symmetric graph]] || Yes || | |||
|- | |||
| [[satisfies property::distance-transitive graph]] || Yes || | |||
|- | |||
| [[satisfies property::bridgeless graph]] || Yes || | |||
|- | |||
| [[dissatisfies property::strongly regular graph]] || No || | |||
|} | |} | ||
Revision as of 04:05, 29 May 2012
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, refers to 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.
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 | |
| size of edge set | 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 | |
| eccentricity of a vertex | 3 |
Other numerical invariants
| Function | Value | Explanation |
|---|---|---|
| clique number | 2 | |
| independence number | 4 | |
| chromatic number | 2 | |
| 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 | |
| girth of a graph | 4 |
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 |