Edge and vertex spaces

In the mathematical discipline of graph theory, the edge space and vertex space of an undirected graph are vector spaces defined in terms of the edge and vertex sets, respectively. These vector spaces make it possible to use techniques of linear algebra in studying the graph.
Definition.
Let formula_1 be a finite undirected graph. The vertex space formula_2 of "G" is the vector space over the finite field of two elements 
formula_3 of all functions formula_4. Every element of formula_2 naturally corresponds the subset of "V" which assigns a 1 to its vertices. Also every subset of "V" is uniquely represented in formula_2 by its characteristic function. The edge space formula_7 is the formula_8-vector space freely generated by the edge set "E". The dimension of the vertex space is thus the number of vertices of the graph, while the dimension of the edge space is the number of edges.
These definitions can be made more explicit. For example, we can describe the edge space as follows:
The singleton subsets of "E" form a basis for formula_7.
One can also think of formula_2 as the power set of "V" made into a vector space with similar vector addition and scalar multiplication as defined for formula_7.
Properties.
The incidence matrix formula_17 for a graph formula_18 defines one possible linear transformation
between the edge space and the vertex space of formula_18. The incidence matrix of formula_18, as a linear transformation, maps each edge to its two incident vertices. Let formula_22 be the edge between formula_23 and formula_24 then
The cycle space and the cut space are linear subspaces of the edge space.