Cut Space of Vertices without Orthogonal Complement of Cycle Space?

Question

I am studying sparse graphs where their complements tend to be dense (not sparse). I understand this so that the sparse graph has a sparse adjacency matrix while its graph complement is not most likely not sparse (exceptions such as $C_5$).

The cut space for undirected graph is defined with the orthogonal complement of the cycle space (linear algebra), not in terms of the complement of graph, inverse of graph (graph theory).

So a question arises:

Does there exist any technique to calculate the cut space without calculating the orthogonal complement of the cycle space?

Helpers

1. What is the orthogonal complement of cycle space for a graph?

2. What is the cut space of vertices? Is it unique?

3. Where to find material to explain graph theory in terms of linear algebra?

Answer 1

Chris made the key observation that to differentiate the graph complement and the orthogonal complement, graph theory versus linearal algebra. The facts that cycle space is the orthogonal complement of the cut space and the cut space is the orthogonal complement of the cycle space is proved in Theorem 1.9.5, Graph Theory by Diestel.

Helpers

3. On Graph theory explained with linear algebra: the observation between orthogonal complement (linear algebra) and complement of graph (graph theory) lead to realise the need to explain graph theory in terms of linear algebra so Textbook on Graph Theory using Linear Algebra and Linear Algebra and Graph theory.