Let P : u0,u1,...,ut be a longest path in a tree T. Show, for every vertex u in T , that ecc(u) = max{dist(u, u0 ), dist(u, ut )}.

To show that for every vertex \( u \) in a tree \( T \), the eccentricity \( \text{ecc}(u) \) is equal to \( \max\{\text{dist}(u, u_0), \text{dist}(u, u_t)\} \), where \( P : u_0, u_1, \ldots, u_t \) is a longest path in \( T \), we proceed as follows:

### Definitions and Setup:
- The **eccentricity** of a vertex \( u \), denoted \( \text{ecc}(u) \), is defined as the greatest distance from \( u \) to any other vertex in the tree \( T \).
- A **longest path** in a tree \( T \) is a path of maximum length (number of edges), and in a tree, such a path is unique up to the choice of endpoints.
- Let \( P : u_0, u_1, \ldots, u_t \) be a longest path in \( T \), with \( u_0 \) and \( u_t \) being the endpoints.

### Proof:
1. **Distance in a Tree**:
   - In a tree, there is exactly one simple path between any two vertices. Thus, the distance \( \text{dist}(u, v) \) is uniquely defined as the length of the path between \( u \) and \( v \).

2. **Eccentricity and Longest Path**:
   - For any vertex \( u \) in \( T \), its eccentricity \( \text{ecc}(u) \) is the maximum distance from \( u \) to any vertex in \( T \).
   - Since \( P \) is the longest path in \( T \), every vertex in \( T \) will lie within the distance from \( u_0 \) to \( u_t \) along this path or be closer to some vertex on this path.

3. **Considering \( u \) on the Longest Path**:
   - If \( u \) is on the path \( P \), say \( u = u_i \) for some \( i \), then the farthest vertices from \( u \) are either \( u_0 \) or \( u_t \). Hence,
     \[
     \text{ecc}(u_i) = \max \{\text{dist}(u_i, u_0), \text{dist}(u_i, u_t)\}.
     \]

4. **Considering \( u \) off the Longest Path**:
   - Suppose \( u \) is not on the path \( P \). Any such \( u \) would have a unique path connecting it to some vertex on \( P \). Let the closest vertex on \( P \) to \( u \) be \( u_i \). The eccentricity \( \text{ecc}(u) \) is then influenced by how far \( u_i \) is from \( u_0 \) and \( u_t \). Therefore,
     \[
     \text{ecc}(u) = \max \{\text{dist}(u, u_0), \text{dist}(u, u_t)\}.
     \]

### Conclusion:
By considering both cases (when \( u \) is on \( P \) and when \( u \) is off \( P \)), we conclude that for every vertex \( u \) in \( T \), the eccentricity \( \text{ecc}(u) \) is given by
\[
\text{ecc}(u) = \max \{\text{dist}(u, u_0), \text{dist}(u, u_t)\}.
\]

This completes the proof.

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Do you want details or have any questions? Here are some additional questions you might find interesting:

1. What is the definition of a tree in graph theory?
2. How can we find the diameter of a tree?
3. What properties differentiate a tree from a general graph?
4. How is the center of a tree determined?
5. What are some applications of trees in computer science?

**Tip:** Understanding the unique path properties in trees can significantly simplify many proofs and algorithms involving trees.

Explore the proof and explanation for the relationship between eccentricity and longest paths in trees. This problem delves into graph theory concepts, including trees, paths, eccentricity, and distance in trees. Suitable for advanced undergraduate students and researchers in graph theory.

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