Question

I want define new graph in sage. Let V be vector space over finite field GF(q). The graph’s vertices are i-dimensional subspace from V and n-i -dimensional subspace from V and two vertices are adjacent if and only if direct sum of two subspace is V.

I have trouble with define this graph in sage. Any suggestion?

Answer 1

This should get you started:

sage: p = 5
sage: K = GF(p^2, 'a')
sage: V = K^4
sage: len(list(V.subspaces(1)))
16276
sage: len(list(V.subspaces(3)))
16276

So this graph is going to be pretty large: 16276 * 2 = 32552 vertices. Let’s do a smaller example. Then you could do something like

sage: p = 3
sage: K = GF(p)
sage: V = K^4
sage: vertices = list(V.subspaces(1)) + list(V.subspaces(3))
sage: for X in vertices:
....:     L = []
....:     for Y in vertices:
....:         if X + Y == V:
....:             L.append(Y)
....:     d[X] = L
....:     
sage: Graph(d)
Graph on 80 vertices