http://www1.tip.nl/~t515027/hypercube.html
Hypercube Computer Game
Theorem 1: If v is an eigenvector of Qn-1 with eigenvalue x then the concatenated vectors [v,v] and [v,-v] are eigenvectors of Qn with eigenvalues x+1 and x-1 respectively.
Eigenvectors of the Adjacency Matrix
=
= +1
11
= -1
1-1
= +2
1111
= 0
11-1-1
= 0
1-11-1
= -2
1-1-11
= +3
11111111
= +1
1111-1-1-1-1
= +1
11-1-111-1-1
= -1
11-1-1-1-111
= +1
1-11-11-11-1
= -1
1-11-1-11-11
= -1
1-1-111-1-11
= -3
1-1-11-111-1
+1
+1
+1 +1 +1 +1
+1
-1
-1
-1-1
-1
-1 -1
n=0
n=1
n=2
n=3
Generating Eigenvectors and Eigenvalues
1
1
1
1
-1
-1
-1
-1
000
001
010
011
100
101
110
111
eigenvector binary number
000
100 110
010
111000
001 011
x
y
z
Theorem 3: Let k be the number of +1 choices in the recursive construction of the eigenvectors of the n-cube. Then for k not equal to n each Walsh state has 2n-k-1 non adjacent subcubes of dimension k that are labeled +1 on their vertices, and 2n-k-1 non adjacent subcubes of dimension k that are labeled -1 on their vertices. If k = n then all the vertices are labeled +1. (Note: Here, "non adjacent" means the subcubes do not share any edges or vertices and there are no edges between the subcubes).
n=5k=3
reduced graph
n=5k=2
reduced graph
Schamtice of the 5-cube Schamtice of the 5-cube
n=5, k= 3 n=5, k= 2