PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: High Entry average rating: Very high
graph topology (Definition)

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 $ \vert G\vert$ of $ G$.

Viewing a graph as a topological space has several advantages:



Anyone with an account can edit this entry. Please help improve it!

"graph topology" is owned by mps. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

See Also: graph theory, graph, connected graph, quotient space, realization of a graph

Log in to rate this entry.
(view current ratings)

Cross-references: fundamental group, tree, connected graph, cell, graph isomorphism, simplicial complexes, interval, unit, topology, edges, vertices, graph
There are 3 references to this entry.

This is version 6 of graph topology, born on 2003-05-08, modified 2006-09-07.
Object id is 4250, canonical name is GraphTopology.
Accessed 5521 times total.

Classification:
AMS MSC05C10 (Combinatorics :: Graph theory :: Topological graph theory, imbedding)
 05C62 (Combinatorics :: Graph theory :: Graph representations )
 54H99 (General topology :: Connections with other structures, applications :: Miscellaneous)

Pending Errata and Addenda
None.
[ View all 4 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)