Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 14 - Graph Theory - 14.4 Trees - Exercise Set 14.4: 29

Answer

The minimum spanning tree includes the nine edges AF, BF, DE, CF, and CE. The total weight of the minimum spanning tree is 91.

Work Step by Step

We can use Kruskal's Algorithm to find the minimum spanning tree for the weighted graph. First, we choose the smallest weight, which is 12. We add the edge AF to the spanning tree. The next smallest weight is 14, so we add edge BF to the spanning tree. The next smallest weight is 16. However, there are two edges with a weight of 16, so we can simply choose one of the two edges arbitrarily. We can add edge DE to the spanning tree. The next smallest weight is 16. However, this edge would make a circuit so we do not add the edge AB to the spanning tree. The next smallest weight is 22, so we add edge CF to the spanning tree. The next smallest weight is 25. However, this edge would make a circuit so we do not add the edge AC to the spanning tree. The next smallest weight is 27, so we add edge CE to the spanning tree. Now we have created a spanning tree that includes all the vertices, is connected, and does not have any circuits. The minimum spanning tree includes the five edges AF, BF, DE, CF, and CE. We can find the total weight of the minimum spanning tree. total weight = 12 + 14 + 16 + 22 + 27 total weight = 91 The total weight of the minimum spanning tree is 91.
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