Graph (topology)

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

A subgraph ${\displaystyle T\subseteq X}$ 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.

• 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 ${\displaystyle X}$ contains at least one maximal tree ${\displaystyle T\subseteq X}$, that is, a tree that is maximal with respect to the order induced by set inclusion on the subgraphs of ${\displaystyle X}$ which are trees.[1]
• If ${\displaystyle X}$ is a graph and ${\displaystyle T\subseteq X}$ a maximal tree, then the fundamental group ${\displaystyle \pi _{1}(X)}$ equals the free group generated by elements ${\displaystyle (f_{\alpha })_{\alpha \in A}}$, where the ${\displaystyle \{f_{\alpha }\}}$ correspond bijectively to the edges of ${\displaystyle X\setminus T}$; in fact, ${\displaystyle X}$ 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]

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