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, where is the 0-skeleton (consisting one point for each vertex ) and are the intervals ("closed one-dimensional unit balls") glued to it, one for each edge .
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.
- Every connected graph contains a maximal tree , that is, a tree that is maximal with respect to the order induced by set inclusion on the subgraphs of which are trees.
- If is a graph and a maximal tree, then the fundamental group equals the free group generated by elements , where the correspond bijectively to the edges of ; in fact, is homotopy equivalent to a wedge sum of circles.
- 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.
- 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 covering space projecting to a graph is also a graph.