centered tree

{{Short description|Tree graph with only one center}}

{{Use American English|date=March 2021}}

{{Use mdy dates|date=March 2021}}

[[Image:Centered tree.gif|right|frame|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 {{mvar|v}} is defined as the greatest distance from {{mvar|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, {{harvtxt|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.{{harv|Harary|1969}}, Theorem 4.2