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: 36

Answer

The maximum spanning tree includes the nine edges BC, CE, AB, DE, CF, EI, FJ, EH, and GH. The total weight of the maximum spanning tree is 92.

Work Step by Step

Normally we use Kruskal's Algorithm to find the minimum spanning tree for a weighted graph. However, we can use a modification of Kruskal's Algorithm to find the maximum spanning tree for a weighted graph. Instead of choosing the edge with the smallest weight at each step, we can choose the edge with the largest weight. First, we choose the largest weight, which is 12. There are two edges with a weight of 12, so we can choose one arbitrarily. We can add the edge BC to the spanning tree. The next largest weight is 12, so we add edge CE to the spanning tree. The next largest weight is 11. There are two edges with a weight of 11, so we can choose one arbitrarily. We can add the edge AB to the spanning tree. The next largest weight is 11, so we add edge DE to the spanning tree. The next largest weight is 10. There are three edges with a weight of 10, so we can choose one arbitrarily. We can add the edge CF to the spanning tree. There are still two edges with a weight of 10, so we can choose one arbitrarily. We can add the edge EI to the spanning tree. The next largest weight is 10. However, edge BE would make a circuit, so we do not add this edge to the spanning tree. The next largest weight is 9. There are three edges with a weight of 9, so we can choose one arbitrarily. We can add the edge FJ to the spanning tree. The next largest weight is 9, so we can add edge EH to the spanning tree. We do not add edge IJ because it would make a circuit. The next largest weight is 8. We can add edge GH 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 maximum spanning tree includes the nine edges BC, CE, AB, DE, CF, EI, FJ, EH, and GH. We can find the total weight of the maximum spanning tree. total weight = 12 + 12 + 11 + 11 + 10 + 10 + 9 + 9 + 8 total weight = 92 The total weight of the maximum spanning tree is 92.
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