Suppose G is a disconnected graph with no circuits. Let N denote the number of vertices

Chapter 7, Problem 52

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QUESTION:

Suppose G is a disconnected graph with no circuits. Let N denote the number of vertices, M the number of edges, and K the number of components. Explain why M = N - K. (Hint: Try Exercise 51 first.)

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QUESTION:

Suppose G is a disconnected graph with no circuits. Let N denote the number of vertices, M the number of edges, and K the number of components. Explain why M = N - K. (Hint: Try Exercise 51 first.)

ANSWER:

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A disconnected graph () is one without any circuits and at least two components. The number of components is , the number of edges is , and the number of vertices is . Since there are no circuits in , each component must be a tree. In light of the fact that a component  is a tree if it has  edges and  vertices.

 

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