Edge and vertex spaces

In the mathematical discipline of graph theory the edge space for a finite undirected edge labeled graph is vector space structure on the edge set of the graph, making it possible to use linear algebra for studying the graph.

Definition

Given a finite undirected edge labeled graph $G:=(V,E)$ with $n$ edges, the edge space ${\mathcal {E}}(G)$ is a $n$-dimensional vector space over $\mathbb {Z} _{2}$. The elements of the vector space are linear combination of edges of $G$ with addition defined as the symmetric difference.

The set of edges $\lbrace \lbrace e_{1}\rbrace ,\ldots ,\lbrace e_{n}\rbrace \rbrace$ forms a canonical basis.

Properties

The incidence matrix $H$ for a graph $G$ defines a linear transformation

$H:{\mathcal {E}}(G)\to {\mathcal {V}}(G)$

between the edge space and the vertex space of $G$.

See also

• vertex space