   Chapter 10.6, Problem 25ES ### 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 if G is a connected, weighted graph and e is an edge of G (not a loop) that has smaller weight than any other edge of G, then e is in every minimum spanning tree for G.

To determine

To prove:

The edge e is in every minimum spanning tree for G ,

where G is a connected, weighted graph and e is an edge of G (not a loop) which has smaller weight than any other edge of G.

Explanation

Given information:

G is connected weighted graph and e is an edge.

Concept used:

Create a new spanning tree T' by adding e to T and removing another edge of T.

Thus, w(T')<w(T).

Proof:

Suppose e is an edge that has smaller weight than any other edge of G and suppose T is a minimum spanning tree for G that does not contain e.

Now create a new spanning tree T' by adding e to T and removing another edge of T

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