Let G be acomplete undirected graph on 4 vertices,having 6 edges with weights being1,2, 3, 4,5,and 6.The maximum possible weight that a minimum weight spanning treeof G can have is _________.a)6.0 : 6.0b)7.0 :7.0c)8.0 : 8.0d)9.0 : 9.0Correct answer is '7'. Can you explain this answer?

Question

Answer

Question Analysis:
The given problem asks us to determine the maximum possible weight that a minimum weight spanning tree of a complete undirected graph on 4 vertices can have. The graph has 6 edges with weights 1, 2, 3, 4, 5, and 6.

Understanding Minimum Weight Spanning Tree (MST):
A minimum weight spanning tree is a tree that connects all the vertices of a graph with the minimum total weight possible. In other words, it is a subgraph of the original graph that is a tree and has the minimum sum of edge weights.

Approach:
To find the maximum possible weight of a minimum weight spanning tree, we need to consider the edges with the highest weights. Since the graph has 6 edges, we can select 3 of them for the minimum weight spanning tree. We need to select the 3 smallest edges to minimize the total weight. Let's consider the possible combinations of 3 edges and calculate their weights:

- Combination 1: {1, 2, 3}
Total weight = 1 + 2 + 3 = 6

- Combination 2: {1, 2, 4}
Total weight = 1 + 2 + 4 = 7

- Combination 3: {1, 2, 5}
Total weight = 1 + 2 + 5 = 8

- Combination 4: {1, 2, 6}
Total weight = 1 + 2 + 6 = 9

- Combination 5: {1, 3, 4}
Total weight = 1 + 3 + 4 = 8

- Combination 6: {1, 3, 5}
Total weight = 1 + 3 + 5 = 9

- Combination 7: {1, 3, 6}
Total weight = 1 + 3 + 6 = 10

- Combination 8: {1, 4, 5}
Total weight = 1 + 4 + 5 = 10

- Combination 9: {1, 4, 6}
Total weight = 1 + 4 + 6 = 11

- Combination 10: {1, 5, 6}
Total weight = 1 + 5 + 6 = 12

The minimum weight spanning tree will have the smallest total weight among these combinations. Hence, the maximum possible weight of the minimum weight spanning tree for the given graph is 7.

Final Answer:
The maximum possible weight that a minimum weight spanning tree of the given graph can have is 7. (Option b)

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