graph topology

A graph (V,E) is identified by its vertices V={v1,v2,} and its edges E={{vi,vj},{vk,vl},}. A graph also admits a natural topology, called the graph topology, by identifying every edge {vi,vj} 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 complexesMathworldPlanetmath. We can form a simplicial complex G={{v}vV}E. And the desired topological realization of the graph is just the geometric realization |G| of G.

Viewing a graph as a topological spaceMathworldPlanetmath 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