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.^[1]

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.^[1]

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.^[1]

A subgraph $T\subseteq X$ 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 $T\subseteq X$ , that is, a tree that is maximal with respect to the order induced by set inclusion on the subgraphs of $X$ which are trees.^[1]
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.^[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]