   Chapter 10.4, Problem 29ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Prove that every nontrivial tree has at least two vertices of degree 1 by filling in the details and completing the following argument: Let T be a nontrivial tree and let S be the set of all paths from one vertex to another in T. Among all the paths in S, choose a path P with a maximum number of edges. (Why is it possible m find such a P?) What can you say about the initial and final vertices of P? Why?

To determine

To prove:

That there are at least two vertices of degree 1 in every nontrivial tree by determining the path with maximum number of edges and the nature of the initial and final vertices of that path.

Explanation

Given information:

T is a nontrivial tree which as a set S that contains all the paths of the tree. Also. P is the path that covers maximum number of edges.

Proof:

A vertex of degree 1 is called a terminal vertex or a leaf, and a vertex of degree 2 is called an internal vertex.

Regarding the nontrivial tree T, the set S is formed with all possible paths in the tree. A path is said to be a walk between two different vertices without repeated vertices or repeated edges

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