Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 14 - Graph Theory - Chapter Summary, Review, and Test - Review Exercises: 41

Answer

The minimum spanning tree includes the eight edges FG, AC, HI, BE, EF, DI, CD, and CG. The total weight of the minimum spanning tree is 4370.

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 360. We add the edge FG to the spanning tree. The next smallest weight is 450, so we add edge AC to the spanning tree. The next smallest weight is 500. However, there are three edges with a weight of 500, so we can simply choose one of the three edges arbitrarily. We can add edge HI to the spanning tree. There are still two edges with a weight of 500, so we can simply choose one of the two edges arbitrarily. We can add edge BE to the spanning tree. The next smallest weight is 500, so we add edge EF to the spanning tree. The next smallest weight is 610, so we add edge DI to the spanning tree. The next smallest weight is 620, so we add edge CD to the spanning tree. The next smallest weight is 630. However, this edge would make a circuit so we do not add the edge BG to the spanning tree. The next smallest weight is 660. However, this edge would make a circuit so we do not add the edge BF to the spanning tree. The next smallest weight is 830, so we add edge CG 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 eight edges FG, AC, HI, BE, EF, DI, CD, and CG. We can find the total weight of the minimum spanning tree. total weight = 360 + 450 + 500 + 500 + 500 + 610 + 620 + 830 total weight = 4370 The total weight of the minimum spanning tree is 4370.
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