Graph (topology)


In topology, a subject in mathematics, a graph is a topological space which arises from a usual graph by replacing vertices by points and each edge by a copy of the unit interval, where is identified with the point associated to and with the point associated to. That is, as topological spaces, graphs are exactly the simplicial 1-complexes and also exactly the one-dimensional CW complexes.
Thus, in particular, it bears the quotient topology of the set
under the quotient map used for gluing. Here is the 0-skeleton, are the intervals glued to it, one for each edge, and is the disjoint union.
The topology on this space is called the graph topology.

Subgraphs and -trees

A subgraph of a graph is a subspace which is also a graph and whose nodes are all contained in the 0-skeleton of. is a subgraph if and only if it consists of vertices and edges from and is closed.
A subgraph is called a tree iff it is contractible as a topological space.

Properties

Using the above properties of graphs, one can prove the Nielsen–Schreier theorem.