Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
### Problem 2: Weighted Graph Analysis **Task:** Consider the following weighted graph. **Graph Details:** - **Vertices:** There are six vertices in the graph, labeled A, B, C, D, E, and F. - **Edges and Weights:** - A to B: 7 - A to C: 6 - A to D: 8 - A to E: 9 - A to F: 8 - B to C: 2 - B to D: 1 - B to E: 3 - B to F: 7 - C to D: 1 - C to E: 5 - C to F: 15 - D to E: 10 - D to F: 12 - E to F: 2 - G to A: 15 - G to B: 4 - G to C: 5 - G to D: 17 - G to E: 13 - G to F: 8 **Instructions:** Use **Kruskal's Algorithm** to determine the minimum cost spanning tree for this graph. Then, draw the tree and provide its cost. **Explaining Kruskal’s Algorithm:** 1. **Sort** all the edges in non-decreasing order of their weight. 2. **Initialize** the minimum spanning tree with no edges. 3. **Add edges** to the minimum spanning tree, one by one, using the sorted list, making sure no cycles are formed. 4. Stop when there are exactly \( V-1 \) edges in the tree, where \( V \) is the number of vertices in the graph. **Purpose:** This exercise will help in understanding how Kruskal’s algorithm can be applied for finding a minimum spanning tree in a graph, which is useful in network design, such as designing least cost networks.
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Transcribed Image Text:### Problem 2: Weighted Graph Analysis **Task:** Consider the following weighted graph. **Graph Details:** - **Vertices:** There are six vertices in the graph, labeled A, B, C, D, E, and F. - **Edges and Weights:** - A to B: 7 - A to C: 6 - A to D: 8 - A to E: 9 - A to F: 8 - B to C: 2 - B to D: 1 - B to E: 3 - B to F: 7 - C to D: 1 - C to E: 5 - C to F: 15 - D to E: 10 - D to F: 12 - E to F: 2 - G to A: 15 - G to B: 4 - G to C: 5 - G to D: 17 - G to E: 13 - G to F: 8 **Instructions:** Use **Kruskal's Algorithm** to determine the minimum cost spanning tree for this graph. Then, draw the tree and provide its cost. **Explaining Kruskal’s Algorithm:** 1. **Sort** all the edges in non-decreasing order of their weight. 2. **Initialize** the minimum spanning tree with no edges. 3. **Add edges** to the minimum spanning tree, one by one, using the sorted list, making sure no cycles are formed. 4. Stop when there are exactly \( V-1 \) edges in the tree, where \( V \) is the number of vertices in the graph. **Purpose:** This exercise will help in understanding how Kruskal’s algorithm can be applied for finding a minimum spanning tree in a graph, which is useful in network design, such as designing least cost networks.
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