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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
)
{\displaystyle G:=(V,E)}
n
{\displaystyle n}
edge space
E
(
G
)
{\displaystyle {\mathcal {E}}(G)}
n
{\displaystyle n}
Z
2
:=
{
0
,
1
}
{\displaystyle \mathbb {Z} _{2}:=\lbrace 0,1\rbrace }
elements of the vector space are subsets of the power set of
E
{\displaystyle E}
vector addition is defined as the symmetric difference :
U
+
V
:=
U
△
V
U
,
V
∈
E
(
G
)
{\displaystyle U+V:=U\triangle V\qquad U,V\in {\mathcal {E}}(G)}
0
⋅
U
:=
∅
U
∈
E
(
G
)
{\displaystyle 0\cdot U:=\emptyset \qquad U\in {\mathcal {E}}(G)}
1
⋅
U
:=
U
U
∈
E
(
G
)
{\displaystyle 1\cdot U:=U\qquad U\in {\mathcal {E}}(G)}
The set of edges
{
{
e
1
}
,
…
,
{
e
n
}
}
{\displaystyle \lbrace \lbrace e_{1}\rbrace ,\ldots ,\lbrace e_{n}\rbrace \rbrace }
E
(
G
)
{\displaystyle {\mathcal {E}}(G)}
Properties
The incidence matrix
H
{\displaystyle H}
G
{\displaystyle G}
linear transformation
H
:
E
(
G
)
→
V
(
G
)
{\displaystyle H:{\mathcal {E}}(G)\to {\mathcal {V}}(G)}
between the edge space and the vertex space of
G
{\displaystyle G}
incident vertices. Let
v
u
{\displaystyle vu}
v
{\displaystyle v}
u
{\displaystyle u}
H
(
v
u
)
=
v
+
u
{\displaystyle H(vu)=v+u}
See also
