Consider the following undirected graph with edge weights as shown: The number…

Key observation: the edges of weight 0.1 form exactly three connected components in the graph.

Reasoning:

• Any minimum spanning tree must use all available edges of weight 0.1 that do not create cycles within their component, because 0.1 is strictly smaller than 0.9.

• After taking the 0.1 edges, the graph splits into three connected components, so we must add exactly two edges to connect these three components into a spanning tree.

• The only available edges that connect these components have weight 0.9, and there are exactly three such inter-component edges, each connecting a different pair of components.

• To obtain a spanning tree we must choose any two of these three inter-component edges; any two chosen will connect all three components and not form a cycle.

Count:

1. Number of ways to pick two edges out of the three inter-component edges = 3.

Conclusion: There are 3 minimum-weight spanning trees.