Consider the following undirected graph with edge weights as shown. The number of minimum-weight spanning trees of the graph is \(\underline{\hspace{2cm}}\).
Show Hint
When multiple edges have the same minimum weight, more than one minimum spanning tree may exist.
Step 1: Identify edges with minimum weight.
From the given graph, the minimum edge weight is \( 0.1 \). All such edges must be considered first while forming a minimum spanning tree.
Step 2: Apply the properties of Minimum Spanning Trees.
A minimum spanning tree connects all vertices with minimum total weight and without forming cycles. Multiple MSTs can exist if there are multiple choices of edges with the same minimum weight.
Step 3: Count valid spanning trees.
By carefully selecting combinations of edges with weight \( 0.1 \) that connect all vertices without cycles, exactly \( 3 \) distinct minimum-weight spanning trees can be formed.
Step 4: Final result.
Hence, the number of minimum-weight spanning trees is \( 3 \).
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