## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 14 - Graph Theory - 14.4 Trees - Exercise Set 14.4: 35

#### Answer

The maximum spanning tree includes the nine edges HI, FJ, EH, CF, BC, FG, BE, AE, and DE. The total weight of the maximum spanning tree is 141.

#### 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 22. We add the edge HI to the spanning tree. The next largest weight is 19, so we add edge FJ to the spanning tree. The next largest weight is 17. However, there are two edges with a weight of 17, so we can simply choose one of the two edges arbitrarily. We can add edge EH to the spanning tree. The next largest weight is 17, so we add edge CF to the spanning tree. The next largest 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 BC to the spanning tree. The next largest weight is 16, so we add edge FG to the spanning tree. The next largest weight is 15. However, there are two edges with a weight of 15, so we can simply choose one of the two edges arbitrarily. We can add edge BE to the spanning tree. The next largest weights are 15, 14, 13, 12, and 11. However, these edges would make a circuit so we do not add these edges to the spanning tree. The next largest weight is 10, so we add edge AE to the spanning tree. The next largest weight is 9, so we add edge DE 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 HI, FJ, EH, CF, BC, FG, BE, AE, and DE. We can find the total weight of the maximum spanning tree. total weight = 22 + 19 + 17 + 17 + 16 + 16 + 15 + 10 + 9 total weight = 141 The total weight of the maximum spanning tree is 141.

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