Centered tree

Short description: Tree graph with 1 or 2 centers

On the left a centered tree, on the right a bicentered one. The numbers show each node's eccentricity.

In the mathematical subfield of graph theory, a centered tree is a tree with only one center, and a bicentered tree is a tree with two centers.

Given a graph, the eccentricity of a vertex v is defined as the greatest distance from v to any other vertex. A center of a graph is a vertex with minimal eccentricity. A graph can have an arbitrary number of centers. However, (Jordan 1869) has proved that for trees, there are only two possibilities:

1. The tree has precisely one center (centered trees).
2. The tree has precisely two centers (bicentered trees). In this case, the two centers are adjacent.

A proof of this fact is given, for example, by Harary.

References

• Jordan, Camille (1869). "Sur les assemblages de lignes" (in French). Journal für die reine und angewandte Mathematik 70 (2): 185–190. http://resolver.sub.uni-goettingen.de/purl?GDZPPN002153998. 
• Harary (1969). Graph Theory. Addison-Wesley Professional. 

External links

• Weisstein, Eric W.. "Bicentered Tree". http://mathworld.wolfram.com/BicenteredTree.html. 
• Weisstein, Eric W.. "Centered Tree". http://mathworld.wolfram.com/CenteredTree.html.