Graph (topology)

Short description: Topological space arising from a usual graph

In topology, a branch of mathematics, a graph is a topological space which arises from a usual graph G=(E,V) by replacing vertices by points and each edge e=xy∈E by a copy of the unit interval I=[0,1], where 0 is identified with the point associated to x and 1 with the point associated to y. That is, as topological spaces, graphs are exactly the simplicial 1-complexes and also exactly the one-dimensional CW complexes.[1]

Thus, in particular, it bears the quotient topology of the set

X0⊔⨆e∈EIe

under the quotient map used for gluing. Here X0 is the 0-skeleton (consisting of one point for each vertex x∈V), Ie are the closed intervals glued to it, one for each edge e∈E, and ⊔ is the disjoint union.[1]

The topology on this space is called the graph topology.

## Contents

* 1 Subgraphs and trees
* 2 Properties
* 3 See also
* 4 References

## Subgraphs and trees

A subgraph of a graph X is a subspace Y⊆X which is also a graph and whose nodes are all contained in the 0-skeleton of X. Y is a subgraph if and only if it consists of vertices and edges from X and is closed.[1]

A subgraph T⊆X is called a tree if it is contractible as a topological space.[1] This can be shown equivalent to the usual definition of a tree in graph theory, namely a connected graph without cycles.

## Properties

* The associated topological space of a graph is connected (with respect to the graph topology) if and only if the original graph is connected.
* Every connected graph X contains at least one maximal tree T⊆X, that is, a tree that is maximal with respect to the order induced by set inclusion on the subgraphs of X which are trees.[1]
* If X is a graph and T⊆X a maximal tree, then the fundamental group π1(X) equals the free group generated by elements (fα)α∈A, where the {fα} correspond bijectively to the edges of X∖T; in fact, X is homotopy equivalent to a wedge sum of circles.[1]
* Forming the topological space associated to a graph as above amounts to a functor from the category of graphs to the category of topological spaces.
* Every covering space projecting to a graph is also a graph.[1]

## See also

* Graph homology
* Topological graph theory
* Nielsen–Schreier theorem, whose standard proof makes use of this concept.

## References

1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. p. 83ff.. ISBN 0-521-79540-0.

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