
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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(Discrete Math)

Transcribed Image Text:### Prim's Algorithm and Minimum Spanning Tree
To find a minimum spanning tree for a weighted undirected graph using Prim's algorithm, you can follow these steps:
1. **Start from the selected vertex (v0)**:
- In this case, vertex \( v_0 \) is the starting point.
2. **Grow the spanning tree**:
- Always choose the smallest weight edge that connects a vertex in the growing spanning tree to a vertex outside of it but is connected to it.
The provided graph and algorithm details are as follows:
#### Graph Breakdown
Vertices: \( v_0, v_1, v_2, v_3, v_4, v_5, v_6, v_7 \)
Edges with Weights:
- \( v_0 - v_5 \) with weight 4
- \( v_0 - v_1 \) with weight 12
- \( v_1 - v_3 \) with weight 7
- \( v_1 - v_2 \) with weight 20
- \( v_2 - v_3 \) with weight 2
- \( v_3 - v_4 \) with weight 15
- \( v_3 - v_7 \) with weight 18
- \( v_4 - v_5 \) with weight 10
- \( v_4 - v_7 \) with weight 15
- \( v_6 - v_7 \) with weight 13
- \( v_5 - v_6 \) with weight 8
#### Steps to Find the Minimum Spanning Tree (MST):
1. Start with vertex \( v_0 \).
2. Select the edge with the smallest weight connected to it (in this case, \( (v_0, v_5) \) with weight 4).
3. Now consider vertices \( v_0, v_5 \).
4. Choose the edge \( (v_5, v_6) \) with weight 8.
5. Now consider vertices \( v_0, v_5, v_6 \).
6. Select the edge \( (v_4, v_5) \) with weight 10.
7. Now consider vertices \( v_0, v_4, v_5, v_6 \).
8. Select the edge \( (v
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