A computer network consists of six computers. Each computer...

Name: Problem 39, Chapter 6: Discrete Mathematics and its Applications
Uploaded: 2019-11-03T05:56:46-08:00
Duration: 1 min 24 s
Channel: Clarissa Noh
Description: 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.]

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.]

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Key Concepts

Degree of a Vertex

The degree of a vertex in a graph is the count of edges incident to that vertex. In a computer network, this represents the number of direct connections a computer has with other computers. Understanding vertex degrees helps reveal important properties of the network, such as the number of direct interactions each computer maintains with others.

Pigeonhole Principle

The pigeonhole principle is a basic principle in combinatorics that states if there are more items than containers, then at least one container must contain more than one item. In the context of this problem, since the possible range of connection counts (degrees) for computers is limited by inherent constraints of the network (i.e., not being able to have both zero and full connectivity simultaneously), the principle guarantees that at least two computers must share the same degree of connectivity.

Graph Theory

Graph theory is the mathematical study of graphs, which are structures used to model pairwise relations between objects. In the context of a computer network, each computer is represented as a vertex, while the direct connections between computers are represented as edges. This abstraction allows the application of graph theoretic principles to analyze and solve problems concerning connectivity and network structure.

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