## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 14 - Graph Theory - 14.4 Trees - Exercise Set 14.4: 27

#### Answer

The minimum spanning tree includes the nine edges DH, AD, AB, EF, DE, GJ, FI, CG, and IJ. The total weight of the minimum spanning tree is 85.

#### 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 4. We add the edge DH to the spanning tree. The next smallest weight is 5, so we add edge AD to the spanning tree. The next smallest weight is 7. However, there are two edges with a weight of 7, so we can simply choose one of the two edges arbitrarily. We can add edge AB to the spanning tree. The next smallest weight is 7, so we add edge EF to the spanning tree. The next smallest weight is 9, so we add edge DE to the spanning tree. The next smallest weight is 10. However, this edge would make a circuit so we do not add the edge AE to the spanning tree. The next smallest weight is 11, so we add edge GJ to the spanning tree. The next smallest weight is 12. However, this edge would make a circuit so we do not add the edge BF to the spanning tree. The next smallest weight is 13, so we add edge FI to the spanning tree. The next smallest weight is 14. However, the edge EI would make a circuit so we do not add the edge EI to the spanning tree. The edge CG also has a weight of 14, so we add edge CG to the spanning tree. The next smallest weight is 15, so we add edge IJ to the spanning tree. The minimum spanning tree includes the nine edges DH, AD, AB, EF, DE, GJ, FI, CG, and IJ. We can find the total weight of the minimum spanning tree. total weight = 4 + 5 + 7 + 7 + 9 + 11 + 13 + 14 + 15 total weight = 85 The total weight of the minimum spanning tree is 85.

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