What is the minimum possible weight of a spanning tree T
Group of choices:
1
8
1
3
2
6
2
1
1
7

Question

What is the minimum possible weight of a spanning tree T
Group of choices:
1
8
1
3
2
6
2
1
1
7

Accepted Answer

⚹ To determine the minimum possible weight of a spanning tree $ T $ in a graph with 6 nodes and 9 edges, we can use Kruskal's algorithm or Prim's algorithm. Both algorithms are designed to find the Minimum Spanning Tree (MST) of a graph. Here, I'll use Kruskal's algorithm for simplicity. ⚹ ⚹ **Steps to find the MST using Kruskal's Algorithm:** 1. **Sort all the edges in non-decreasing order of their weight.** 2. **Pick the smallest edge. Check if it forms a cycle with the spanning tree formed so far. If a cycle is not formed, include this edge. Else, discard it.** 3. **Repeat step 2 until there are $ V-1 $ edges in the spanning tree, where $ V $ is the number of vertices.** ⚹ **Given edges and their weights:** - (1, 2) - 2 - (1, 4) - 1 - (1, 5) - 4 - (2, 3) - 3 - (2, 4) - 3 - (2, 6) - 7 - (3, 4) - 5 - (3, 6) - 8 - (4, 5) - 9 ⚹ **Sorted edges by weight:** 1. (1, 4) - 1 2. (1, 2) - 2 3. (2, 3) - 3 4. (2, 4) - 3 5. (1, 5) - 4 6. (3, 4) - 5 7. (2, 6) - 7 8. (3, 6) - 8 9. (4, 5) - 9 ⚹ **Steps to form the MST:** - Pick (1, 4) - 1 - Pick (1, 2) - 2 - Pick (2, 3) - 3 - Pick (2, 4) - 3 (forms a cycle, discard) - Pick (1, 5) - 4 - Pick (3, 4) - 5 (forms a cycle, discard) - Pick (2, 6) - 7 ⚹ **Edges in the MST:** - (1, 4) - 1 - (1, 2) - 2 - (2, 3) - 3 - (1, 5) - 4 - (2, 6) - 7 ⚹ **Total weight of the MST:** $$ 1 + 2 + 3 + 4 + 7 = 17 $$ ⚹ The minimum possible weight of a spanning tree $ T $ is 17. ⚹

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