Centered tree explained

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 is defined as the greatest distance from 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, 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. , Theorem 4.2

References

• Jordan. Camille. Camille Jordan. 1869. Sur les assemblages de lignes. Journal für die reine und angewandte Mathematik. 70. 2. 185–190. French.
• Book: Harary , Frank . Frank Harary

. Graph Theory. Frank Harary. 1969. Addison-Wesley Professional .