# graph topology

A graph $(V,E)$ is identified by its vertices $V=\{v_{1},v_{2},\ldots\}$ and its edges $E=\{\{v_{i},v_{j}\},\{v_{k},v_{l}\},\ldots\}$. A graph also admits a natural topology, called the graph topology, by identifying every edge $\{v_{i},v_{j}\}$ with the unit interval $I=[0,1]$ and gluing them together at coincident vertices.

This construction can be easily realized in the framework of simplicial complexes. We can form a simplicial complex $G=\left\{\{v\}\mid v\in V\right\}\cup E$. And the desired topological realization of the graph is just the geometric realization $|G|$ of $G$.

Viewing a graph as a topological space has several advantages:

Remark: A graph is/can be regarded as a one-dimensional $CW$-complex.

 Title graph topology Canonical name GraphTopology Date of creation 2013-03-22 13:37:03 Last modified on 2013-03-22 13:37:03 Owner mps (409) Last modified by mps (409) Numerical id 10 Author mps (409) Entry type Definition Classification msc 54H99 Classification msc 05C62 Classification msc 05C10 Synonym one-dimensional CW complex Related topic GraphTheory Related topic Graph Related topic ConnectedGraph Related topic QuotientSpace Related topic Realization Related topic RSupercategory Related topic CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams