Question 27

The graph contains 9 vertices arranged in a $3 \times 3$ grid with 12 edges in total. The edge weights are either 0.1 or 0.9.

According to Kruskal's algorithm, we must first include all cycle-free edges of minimum weight 0.1. There are 6 edges of weight 0.1, and together they do not form any cycles. These 6 edges reduce the 9 vertices into 3 connected components.

To connect the 3 components into a single spanning tree of 8 edges, exactly 2 additional edges must be selected from the remaining edges of weight 0.9 without creating a cycle. Examining the cut between the components reveals exactly 3 distinct pairs of weight 0.9 edges that successfully connect the 3 components into a tree.

Hence, the total number of minimum-weight spanning trees is 3.