true or false: the highest weight edge in a graph will never be in an mst. if true, prove it. if false, show an example in which it is included.

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09-12-2022
Mathematics

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true or false: the highest weight edge in a graph will never be in an mst. if true, prove it. if false, show an example in which it is included.

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abhinavjh321 abhinavjh321 · Answer 1 · 2022-12-17T05:36:28-05:00

False. Consider a graph with 4 vertices, where there is an edge of weight 8 between the two farthest vertices, and an edge of weight 10 between the two closest vertices.

The highest weight edge (10) would be included in the MST, as it is the only connection between the two closest vertices.The definition of a minimum spanning tree states that it is a subset of edges of a graph that connect all of the vertices without forming any cycles, and with the minimum total weight. If a graph contains an edge with the highest weight, it may still be included in the MST if it is the only connection between two vertices. To illustrate this, consider a graph with 4 vertices, where there is an edge of weight 8 between the two farthest vertices, and an edge of weight 10 between the two closest vertices. In this case, the highest weight edge (10) would be included in the MST, as it is the only connection between the two closest vertices, and thus the minimum total weight is achieved.

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