The cube graph

The cube graph

has incidence matrix

If is a field, the kernel of is the cycle space. The cycle space has basis

A cut is a partition of the vertex set of into two subsets, . The cocycle of such a cut is the set of edges that have one endpoint in and the other endpoint in . The cocycle space has basis

The Sagemath command below use functions from the file alg-graph-thry1.sage.

sage: Gamma = graphs.CubeGraph(3)
sage: eo = [1]*12
sage: incidence_matrix(Gamma, eo)
[ 1 -1  0  0  0  0  0  0]
[ 1  0 -1  0  0  0  0  0]
[ 1  0  0  0 -1  0  0  0]
[ 0  1  0 -1  0  0  0  0]
[ 0  1  0  0  0 -1  0  0]
[ 0  0  1 -1  0  0  0  0]
[ 0  0  1  0  0  0 -1  0]
[ 0  0  0  1  0  0  0 -1]
[ 0  0  0  0  1 -1  0  0]
[ 0  0  0  0  1  0 -1  0]
[ 0  0  0  0  0  1  0 -1]
[ 0  0  0  0  0  0  1 -1]
sage: cycle_space(Gamma, eo)
Vector space of degree 12 and dimension 5 over Rational Field
Basis matrix:
[ 1  0 -1  0 -1  0  0  0  0 -1 -1  1]
[ 0  1  1  0  0  0 -1  0  0  1  0  0]
[ 0  0  0  1  1  0  0 -1  0  0  1  0]
[ 0  0  0  0  0  1  1  1  0  0  0 -1]
[ 0  0  0  0  0  0  0  0  1 -1 -1  1]
sage: cocycle_space(Gamma, eo)
Vector space of degree 12 and dimension 7 over Rational Field
Basis matrix:
[ 1  0  0  0  0 -1  0  0  1  0  0 -1]
[ 0  1  0  0  0 -1  0  0  0 -1  0 -1]
[ 0  0  1  0  0  0  0  0 -1 -1  0  0]
[ 0  0  0  1  0 -1  0  0  0  0 -1 -1]
[ 0  0  0  0  1  0  0  0 -1  0 -1  0]
[ 0  0  0  0  0  0  1  0  0  1  0  1]
[ 0  0  0  0  0  0  0  1  0  0  1  1]