# $$n$$-Cube#

This section provides some examples on Chapter 2 of Stanley’s book [Stanley2013], which deals with $$n$$-cubes, the Radon transform, and combinatorial formulas for walks on the $$n$$-cube.

The vertices of the $$n$$-cube can be described by vectors in $$\mathbb{Z}_2^n$$. First we define the addition of two vectors $$u,v \in \mathbb{Z}_2^n$$ via the following distance:

sage: def dist(u,v):
....:     h = [(u[i]+v[i])%2 for i in range(len(u))]
....:     return sum(h)

>>> from sage.all import *
>>> def dist(u,v):
...     h = [(u[i]+v[i])%Integer(2) for i in range(len(u))]
...     return sum(h)


The distance function measures in how many slots two vectors in $$\mathbb{Z}_2^n$$ differ:

sage: u = (1,0,1,1,1,0)
sage: v = (0,0,1,1,0,0)
sage: dist(u,v)
2

>>> from sage.all import *
>>> u = (Integer(1),Integer(0),Integer(1),Integer(1),Integer(1),Integer(0))
>>> v = (Integer(0),Integer(0),Integer(1),Integer(1),Integer(0),Integer(0))
>>> dist(u,v)
2


Now we are going to define the $$n$$-cube as the graph with vertices in $$\mathbb{Z}_2^n$$ and edges between vertex $$u$$ and vertex $$v$$ if they differ in one slot, that is, the distance function is 1:

sage: def cube(n):
....:     G = Graph(2**n)
....:     vertices = Tuples([0,1],n)
....:     for i in range(2**n):
....:         for j in range(2**n):
....:             if dist(vertices[i],vertices[j]) == 1:
....:                 G.add_edge(i,j)
....:     return G

>>> from sage.all import *
>>> def cube(n):
...     G = Graph(Integer(2)**n)
...     vertices = Tuples([Integer(0),Integer(1)],n)
...     for i in range(Integer(2)**n):
...         for j in range(Integer(2)**n):
...             if dist(vertices[i],vertices[j]) == Integer(1):
...                 G.add_edge(i,j)
...     return G


We can plot the $$3$$ and $$4$$-cube:

sage: cube(3).plot()
Graphics object consisting of 21 graphics primitives

>>> from sage.all import *
>>> cube(Integer(3)).plot()
Graphics object consisting of 21 graphics primitives

sage: cube(4).plot()
Graphics object consisting of 49 graphics primitives

>>> from sage.all import *
>>> cube(Integer(4)).plot()
Graphics object consisting of 49 graphics primitives


Next we can experiment and check Corollary 2.4 in Stanley’s book, which states the $$n$$-cube has $$n$$ choose $$i$$ eigenvalues equal to $$n-2i$$:

sage: G = cube(2)
sage: G.adjacency_matrix().eigenvalues()
[2, -2, 0, 0]

sage: G = cube(3)
sage: G.adjacency_matrix().eigenvalues()
[3, -3, 1, 1, 1, -1, -1, -1]

sage: G = cube(4)
sage: G.adjacency_matrix().eigenvalues()
[4, -4, 2, 2, 2, 2, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0]

>>> from sage.all import *
>>> G = cube(Integer(2))
>>> G.adjacency_matrix().eigenvalues()
[2, -2, 0, 0]

>>> G = cube(Integer(3))
>>> G.adjacency_matrix().eigenvalues()
[3, -3, 1, 1, 1, -1, -1, -1]

>>> G = cube(Integer(4))
>>> G.adjacency_matrix().eigenvalues()
[4, -4, 2, 2, 2, 2, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0]


It is now easy to slightly vary this problem and change the edge set by connecting vertices $$u$$ and $$v$$ if their distance is 2 (see Problem 4 in Chapter 2):

sage: def cube_2(n):
....:     G = Graph(2**n)
....:     vertices = Tuples([0,1],n)
....:     for i in range(2**n):
....:         for j in range(2**n):
....:             if dist(vertices[i],vertices[j]) == 2:
....:                 G.add_edge(i,j)
....:     return G

sage: G = cube_2(2)
sage: G.adjacency_matrix().eigenvalues()
[1, 1, -1, -1]

sage: G = cube_2(4)
sage: G.adjacency_matrix().eigenvalues()
[6, 6, -2, -2, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0]

>>> from sage.all import *
>>> def cube_2(n):
...     G = Graph(Integer(2)**n)
...     vertices = Tuples([Integer(0),Integer(1)],n)
...     for i in range(Integer(2)**n):
...         for j in range(Integer(2)**n):
...             if dist(vertices[i],vertices[j]) == Integer(2):
...                 G.add_edge(i,j)
...     return G

>>> G = cube_2(Integer(2))
>>> G.adjacency_matrix().eigenvalues()
[1, 1, -1, -1]

>>> G = cube_2(Integer(4))
>>> G.adjacency_matrix().eigenvalues()
[6, 6, -2, -2, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0]


Note that the graph is in fact disconnected. Do you understand why?

sage: cube_2(4).plot()
Graphics object consisting of 65 graphics primitives

>>> from sage.all import *
>>> cube_2(Integer(4)).plot()
Graphics object consisting of 65 graphics primitives