Consider a complete undirected graph with vertex set {0, 1, 2, 3, 4}. Entry Wij in the matrix W below is the weight of the edge {i, j}. What is the minimum possible weight of a spanning tree T in this graph such that vertex 0 is a leaf node in the tree T?Marks : 1Negative Marks : 0Answer here810710
Question
Consider a complete undirected graph with vertex set {0, 1, 2, 3, 4}. Entry Wij in the matrix W below is the weight of the edge {i, j}. What is the minimum possible weight of a spanning tree T in this graph such that vertex 0 is a leaf node in the tree T?Marks : 1Negative Marks : 0Answer here810710
Solution
To find the minimum possible weight of a spanning tree T in the given graph such that vertex 0 is a leaf node, we can use the Prim's algorithm.
- Initialize an empty set S to store the vertices included in the spanning tree T.
- Initialize a priority queue Q to store the edges of the graph, sorted by their weights.
- Start with vertex 0 and add it to S.
- Add all the edges incident to vertex 0 to the priority queue Q.
- Repeat the following steps until all vertices are included in S: a. Remove the edge with the minimum weight from Q. b. If the edge connects a vertex in S to a vertex not in S, add the vertex not in S to S and add all its incident edges to Q.
- The total weight of the spanning tree T is the sum of the weights of all the edges included in T.
In this case, the given graph is a complete undirected graph with vertex set {0, 1, 2, 3, 4}. The weight of the edge {i, j} is given by the entry Wij in the matrix W.
To find the minimum possible weight of a spanning tree T such that vertex 0 is a leaf node, we need to consider the edges incident to vertex 0. Since vertex 0 is a leaf node, it will have only one incident edge.
Therefore, we need to find the minimum weight among the edges incident to vertex 0. In the given matrix W, the weights of the edges incident to vertex 0 are 8, 1, 0, 7, and 10.
The minimum weight among these edges is 0. Therefore, the minimum possible weight of a spanning tree T in this graph such that vertex 0 is a leaf node is 0.
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