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: 42

Answer

The minimum spanning tree includes the nine edges FG, CE, BD, DE, EF, GJ, AB, EI, and HI. The layout of the trail system includes these nine edges. The minimum mileage for the bike trail is 164 miles.

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 2. We add the edge FG to the spanning tree. The next smallest weight is 6, so we add edge CE to the spanning tree. The next smallest weight is 8, so we add edge BD to the spanning tree. The next smallest weight is 15, so we add edge DE to the spanning tree. The next smallest weight is 17, so we add edge EF to the spanning tree. The next smallest weight is 22. However, these three edges would make a circuit so we do not add these edges to the spanning tree. The next smallest weight is 25. However, there are two edges with a weight of 25, so we can simply choose one of the two edges arbitrarily. We can add edge GJ to the spanning tree. The other edge with a weight of 25 would make a circuit so we do not add the edge BC to the spanning tree. The next smallest weight is 26, so we add edge AB to the spanning tree. The next smallest weight is 30, so we add edge EI to the spanning tree. The next smallest weight is 33. However, this edge would make a circuit so we do not add the edge DI to the spanning tree. The next smallest weight is 35. However, there are two edges with a weight of 35, so we can simply choose one of the two edges arbitrarily. We can add edge HI 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 FG, CE, BD, DE, EF, GJ, AB, EI, and HI. The layout of the trail system includes these nine edges. We can find the total weight of the minimum spanning tree. total weight = 2 + 6 + 8 + 15 + 17 + 25 + 26 + 30 + 35 total weight = 164 The total weight of the minimum spanning tree is 164. Since the weights represent miles, the minimum mileage for the bike trail is 164 miles.
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