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  be a finite undirected graph. The vertex space  of G is the vector space over the finite field of two elements 
 of all functions . Every element of  naturally corresponds the subset of V which assigns a 1 to its vertices. Also every subset of V is uniquely represented in  by its characteristic function. The edge space  is the -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:
 elements are subsets of , that is, as a set  is the power set of E
 vector addition is defined as the symmetric difference: 
 scalar multiplication is defined by:
 
 
The singleton subsets of E form a basis for . One can also think of  as the power set of V made into a vector space with similar vector addition and scalar multiplication as defined for . Properties
The incidence matrix  for a graph  defines one possible linear transformation

between the edge space and the vertex space of . The incidence matrix of , as a linear transformation, maps each edge to its two incident vertices. Let  be the edge between  and  then

The cycle space and the cut space are linear subspaces of the edge space. References
 (the electronic 3rd edition is freely available on author's site). See also
cycle space
cut space

Algebraic graph theory