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In a connected weighted graph with vertices, all the edges have distinct positive integer weights. Then, the maximum number of minimum weight spanning trees in the graph is
- a)1
- b)n
- c)equal to number of edges in the graph.
- d)equal to maximum weight of an edge of the graph.
- e)nn2
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
In a connected weighted graph with vertices, all the edges have distin...
There will be unique min weight spanning tree since all weights are distinct.
Most Upvoted Answer
In a connected weighted graph with vertices, all the edges have distin...
The maximum number of minimum weight spanning trees in a connected weighted graph with distinct positive integer weights is 1.
Explanation:
- A minimum weight spanning tree is a tree that connects all the vertices of a graph with the minimum total weight possible.
- The weight of a spanning tree is the sum of the weights of all its edges.
- In a connected graph, a spanning tree is a subgraph that is a tree and contains all the vertices of the graph.
- The minimum weight spanning tree is unique if all the edges have distinct weights.
Proof:
- Let's assume that there are multiple minimum weight spanning trees in the graph.
- Since all the edges have distinct weights, there must be at least one edge in each minimum weight spanning tree that is not present in the other minimum weight spanning trees.
- Let's consider two minimum weight spanning trees T1 and T2 that have at least one edge E1 and E2 respectively, such that E1 is not present in T2 and E2 is not present in T1.
- Now, if we remove E1 from T1 and add E2, and remove E2 from T2 and add E1, we will still have a spanning tree in both cases.
- However, the weight of the new spanning tree formed by adding E2 to T1 and removing E1 will be less than the weight of T1, and similarly, the weight of the new spanning tree formed by adding E1 to T2 and removing E2 will be less than the weight of T2.
- This contradicts the assumption that T1 and T2 are minimum weight spanning trees.
- Therefore, there can be only one minimum weight spanning tree in the graph.
Hence, the maximum number of minimum weight spanning trees in the graph is 1.
Explanation:
- A minimum weight spanning tree is a tree that connects all the vertices of a graph with the minimum total weight possible.
- The weight of a spanning tree is the sum of the weights of all its edges.
- In a connected graph, a spanning tree is a subgraph that is a tree and contains all the vertices of the graph.
- The minimum weight spanning tree is unique if all the edges have distinct weights.
Proof:
- Let's assume that there are multiple minimum weight spanning trees in the graph.
- Since all the edges have distinct weights, there must be at least one edge in each minimum weight spanning tree that is not present in the other minimum weight spanning trees.
- Let's consider two minimum weight spanning trees T1 and T2 that have at least one edge E1 and E2 respectively, such that E1 is not present in T2 and E2 is not present in T1.
- Now, if we remove E1 from T1 and add E2, and remove E2 from T2 and add E1, we will still have a spanning tree in both cases.
- However, the weight of the new spanning tree formed by adding E2 to T1 and removing E1 will be less than the weight of T1, and similarly, the weight of the new spanning tree formed by adding E1 to T2 and removing E2 will be less than the weight of T2.
- This contradicts the assumption that T1 and T2 are minimum weight spanning trees.
- Therefore, there can be only one minimum weight spanning tree in the graph.
Hence, the maximum number of minimum weight spanning trees in the graph is 1.
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In a connected weighted graph with vertices, all the edges have distinct positive integer weights. Then, the maximum number of minimum weight spanning trees in the graph isa)1b)nc)equal to number of edges in the graph.d)equal to maximum weight of an edge of the graph.e)nn2Correct answer is option 'A'. Can you explain this answer?
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