Graph (topology)

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 $$e = xy \in E$$ 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.

Thus, in particular, it bears the quotient topology of the set
 * $$X_0 \sqcup \bigsqcup_{e \in E} I_e$$

under the quotient map used for gluing. Here $$X_0$$ is the 0-skeleton (consisting of one point for each vertex $$x \in V$$), $$I_e$$ are the closed intervals glued to it, one for each edge $$e \in E$$, and $$\sqcup$$ is the disjoint union.

The topology on this space is called the graph topology.

Subgraphs and trees
A subgraph of a graph $$X$$ is a subspace $$Y \subseteq X$$ 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.

A subgraph $$T \subseteq X$$ is called a tree if it is contractible as a topological space. 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 $$T \subseteq X$$, that is, a tree that is maximal with respect to the order induced by set inclusion on the subgraphs of $$X$$ which are trees.
 * If $$X$$ is a graph and $$T \subseteq X$$ a maximal tree, then the fundamental group $$\pi_1(X)$$ equals the free group generated by elements $$(f_\alpha)_{\alpha \in A}$$, where the $$\{f_\alpha\}$$ correspond bijectively to the edges of $$X \setminus T$$; in fact, $$X$$ 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.
 * Every covering space projecting to a graph is also a graph.