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Chapter 6: Problem 39

A computer network consists of six computers. Each computer is directly connected to zero or more of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of other computers. [Hint: It is impossible to have a computer linked to none of the others and a computer linked to all the others.]

Short Answer

Expert verified
At least two computers have the same number of connections (using the pigeonhole principle).

Step by step solution

01

Understand the Problem

There are six computers. We need to show that at least two computers are directly connected to the same number of other computers. Use the pigeonhole principle.
02

Connection Possibilities

Each computer can be connected to 0, 1, 2, 3, 4, or 5 other computers. So, there are 6 possible values.
03

Applying Constraints

According to the hint, it is not possible to have a computer connected to 0 others if there is one connected to 5 others, since they must be connected to each other.
04

Valid Values

Therefore, actual possible connection counts are from 1 to 5, which are 5 possible values.
05

Using the Pigeonhole Principle

There are 6 computers and 5 possible connection counts. By the pigeonhole principle, at least two computers must have the same number of connections.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Computer Networks
A computer network is a set of computers connected to share resources and communicate with each other. In our problem, there are six computers.
Each computer can connect to any number of other computers in the network. Here's how a computer network functions:
  • Devices are interconnected, allowing data transfer and resource sharing.
  • Connections can be physical (like cables) or wireless.
  • Networks improve communication efficiency and resource utilization.

In the exercise, the goal is to understand how these six computers, with varying connections, fit into the concept of the pigeonhole principle.
Making Connections in Networks
Connections in computer networks are crucial for communication between devices.
Here鈥檚 what you need to know about connections in our context:
  • Each computer can connect to 0 to 5 other computers.
  • Different computers can have different numbers of connections.
  • Connection count impacts the data flow and network dynamics.

In this problem, we consider the constraints that prevent a computer from being connected to none if another is connected to all.
Hence, valid connection counts range from 1 to 5.
Discrete Mathematics in Networking
Discrete mathematics deals with distinct and separate values, which is perfect for analyzing networks.
This field helps us understand structures and relationships within networks:
  • Focuses on countable, distinct elements.
  • Includes concepts like graph theory, which is essential for network analysis.
  • Uses mathematical principles such as the pigeonhole principle to solve problems.

In this exercise, discrete mathematics allows us to break down the possible connections and ensure that, by using the pigeonhole principle, certain conditions are met.
Applying Graph Theory
Graph theory is a branch of discrete mathematics that studies graphs, which are mathematical structures used to model pairwise relations between objects.
Here鈥檚 how it applies to our network problem:
  • Each computer is a vertex in the graph.
  • Connections between computers are edges in the graph.
  • Each vertex can have a certain degree, representing the number of connections.

By understanding the vertices and edges' structure, we apply the pigeonhole principle to show that in a network with six computers and five possible connection values, there must be at least two computers with the same degree of connection.

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