Chemistry - Edge space

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

Let $G : = (V, E)$ be a finite undirected edge labeled graph with $n$ edges. The edge space $\mathcal{E}(G)$ is an $n$ dimensional vector space over $\mathbb{Z}_2:=\lbrace 0,1 \rbrace$ defined as follows

elements of the vector space are subsets of the power set of $E$
vector addition is defined as the symmetric difference: $U+V:=U \triangle V \qquad U,V \in \mathcal{E}(G)$
$0 \cdot U := \emptyset \qquad U \in \mathcal{E}(G)$
$1 \cdot U := U \qquad U \in \mathcal{E}(G)$

The set of edges $\lbrace \lbrace e_1 \rbrace,\ldots,\lbrace e_n \rbrace \rbrace$ forms a canonical basis for $\mathcal{E}(G)$ .

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 $G$ . It maps each edge to its two incident vertices. Let $v u$ be the edge between $v$ and $u$ then

H (v u) = v + u

The cycle space and the cut space are linear subspaces of the edge space.

Edge space

Definition

Properties

See also