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Chapter 14: Problem 45

Describe the relationship between the number of vertices and the number of edges in a tree.

Short Answer

Expert verified
The relationship between the number of vertices \(v\) and the number of edges \(e\) in a tree can be described by the equation \(e = v - 1\).

Step by step solution

01

Defining Terms

Define what a tree, vertices, and edges are in terms of graph theory. A 'tree' is a connected acyclic graph; 'vertices' are points where two or more edges meet, and 'edges' are lines between two vertices.
02

Relating Vertices to Edges in a Tree

Understand that in a tree, due to its nature of being acyclic and connected, each vertex is added by creating a new edge. Hence, for a tree with \(v\) vertices, there will be \(v-1\) edges.
03

Proving the Relationship

Assume that a tree starts with one vertex, thus having 0 edges. Each new vertex introduced to the tree adds exactly one new edge because a tree is acyclic (meaning no cycles or loops are present) and remains connected. Therefore, for \(v\) vertices, there will be \(v-1\) edges.
04

Summary of Relationship

The relationship between the number of vertices \(v\) and the number of edges \(e\) in a tree can be described as \(e = v - 1\). This holds true for all trees.

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Most popular questions from this chapter

Why is the Brute Force Method impractical for large numbers of vertices?

Group members should determine a relationship that exists among some, but not all, members. Did some of you know one another before the course began? Do some of you have the same academic major? Be as creative as possible in determining this relationship. Then create a graph that serves as a model for describing this relationship.

In Exercises 15-18, determine the number of Hamilton circuits in a complete graph with the given number of vertices. 12

What is a graph? Define vertices and edges as part of your description.

What is a spanning tree?
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