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:

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

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

• A connected graph is a tree if and only if its fundamental group is trivial.

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