Centered tree

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.^[1]

Notes

1. (Harary 1969), Theorem 4.2

References

• Jordan, Camille (1869). "Sur les assemblages de lignes". Journal für die reine und angewandte Mathematik (in French). 70 (2): 185–190.
• Harary, Frank (1969). Graph Theory. Addison-Wesley Professional.

External links

• Weisstein, Eric W. "Bicentered Tree". MathWorld.
• Weisstein, Eric W. "Centered Tree". MathWorld.