graph topology
A graph $(V,E)$ is identified by its vertices $V=\{{v}_{1},{v}_{2},\mathrm{\dots}\}$ and its edges $E=\{\{{v}_{i},{v}_{j}\},\{{v}_{k},{v}_{l}\},\mathrm{\dots}\}$. 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 $G=\{\{v\}\mid v\in V\}\cup E$. 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:

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The notion of graph isomorphism^{} becomes that of simplicial (or cell) complex (http://planetmath.org/CWComplex) isomorphism^{}.

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The notion of a connected graph^{} coincides with topological connectedness (http://planetmath.org/ConnectedSpace).

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A connected graph is a tree if and only if its fundamental group^{} is trivial.
Remark: A graph is/can be regarded as a onedimensional $CW$complex.
Title  graph topology 
Canonical name  GraphTopology 
Date of creation  20130322 13:37:03 
Last modified on  20130322 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  onedimensional CW complex 
Related topic  GraphTheory 
Related topic  Graph 
Related topic  ConnectedGraph 
Related topic  QuotientSpace 
Related topic  Realization 
Related topic  RSupercategory 
Related topic  CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams 