Centered tree

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:

The tree has precisely one center (centered trees).
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

(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

[1] (Harary 1969), Theorem 4.2

1