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: 28

Answer

The minimum spanning tree includes the nine edges EF, DG, HI, AC, DH, FI, BD, FJ, and CF. The total weight of the minimum spanning tree is 65.

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 5. We add the edge EF to the spanning tree. The next smallest weight is 6. However, there are two edges with a weight of 6, so we can simply choose one of the two edges arbitrarily. We can add edge DG to the spanning tree. The next smallest weight is 6, so we add edge HI to the spanning tree. The next smallest weight is 7. However, there are three edges with a weight of 7, so we can simply choose one of the three edges arbitrarily. We can add edge AC to the spanning tree. There are still two edges with a weight of 7, so we can simply choose one of the two edges arbitrarily. We can add edge DH to the spanning tree. The next smallest weight is 7, so we add edge FI to the spanning tree. The next smallest weight is 8. However, there are two edges with a weight of 8, so we can simply choose one of the two edges arbitrarily. We can add edge BD to the spanning tree. The next smallest weight is 8. However, this edge would make a circuit so we do not add the edge GH to the spanning tree. The next smallest weight is 9. However, there are three edges with a weight of 9, so we can simply choose one of the three edges arbitrarily. We can add edge FJ to the spanning tree. There are still two edges with a weight of 9. However, both edges would make a circuit so we do not add the edge IJ or the edge EH to the spanning tree. The next smallest weight is 10. However, there are three edges with a weight of 10, so we can simply choose one of the three edges arbitrarily. We can add edge CF 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 nine edges EF, DG, HI, AC, DH, FI, BD, FJ, and CF. We can find the total weight of the minimum spanning tree. total weight = 5 + 6 + 6 + 7 + 7 + 7 + 8 + 9 + 10 total weight = 65 The total weight of the minimum spanning tree is 65.
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