Cycle spaces and cocycle spaces of a graph using Sagemath

The cube graph \Gamma

The cube graph

The cube graph

has incidence matrix

B =   \left(\begin{array}{rrrrrrrr}  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  \end{array}\right).

If F is a field, the kernel of B:C^1(\Gamma,F)\to C^0(\Gamma,F) is the cycle space. The cycle space has basis

(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),

A cut is a partition of the vertex set of \Gamma=(V,E) into two subsets, V= S \cup T. The cocycle of such a cut is the set \{(u,v)\in E\ |\ u\in S,\ v \in T\} of edges that have one endpoint in S and the other endpoint in T. The cocycle space has basis

(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).

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]

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s