I'm currently studying out of Dummit and Foote's Graph Theory and Its Applications (2nd ed.), and am having trouble understanding the significance of one of the sections. More specifically, section 4.6 defines a vector space of edge subsets of a graph over the scalar field $GF(2)$. They then consider the subspaces of cycles and edge-cuts in this vector spaces, and define a basis for each space in terms of the fundamental systems of cycles and edge-cuts of spanning trees. This allows them to find the dimension of the cycle and edge-cut spaces and prove that the two spaces are orthogonal to each other.
Establishing links between graph theory and linear algebra seems good and useful. However, the definition of the edge space of a graph fields rather forced, and the book doesn't provide any particularly deep results about it. My question is this: what makes the edge, cycle, and edge-cut vector spaces useful objects of study? Is there some important theorem that hinges on them somehow? Do they motivate an interesting area of study? Sorry if this question is too open-ended, but I've asked my professors and done some online research to no avail.
Best Answer
The vector space of cycles of a graph is the simplest nontrivial case of the homology group of a simplicial complex, so in some sense this vector space motivates about half of algebraic topology.
Back in the realm of graph theory, a cycle basis is relevant to solving many problems whose constraints arise from the cycles of a graph, because a small basis stands in for exponentially many cycles. A simple example of this is the application of Kirchoff's law for the potential differences in an electric circuit.
The law states that around any cycle, the sum of the potential differences is $0$. This gives as many constraints as there are cycles, and in a dense graph, many of them are redundant and it's not clear what the relevant ones are. A cycle basis gives us a small list of sufficient constraints.