Question

The eccentricity of a vertex in an unrooted tree is the length of the longest simple path beginning at this vertex. A vertex is called a center if no vertex in the tree has smaller eccentricity than this vertex. In Exercises 39-41 find every vertex that is a center in the given tree.

40.

Short answer

Here e is the only vertex in the tree having a minimal eccentricity. So, e is the center of the tree.

Step-by-step solution

Definition

The eccentricity of a vertex is the length of the longest simple path starting at the vertex in the unrooted tree.

A vertex is called a center if no vertex within the tree has a smaller eccentricity than this vertex.

Eccentricity of a vertex

In the given tree,

Eccentricity of \(a=6\)

Eccentricity of \(b=5\)

Eccentricity of \(c=6\)

Eccentricity of \(d=4\)

Eccentricity of \(e=3\)

Eccentricity of \(f=4\)

Eccentricity of \(g=5\)

Eccentricity of \(h=5\)

Eccentricity of \(i=6\)

Eccentricity of \(j=6\)

Eccentricity of \(k=6\)