Consider the following undirected graph, where each edge is labeled with its…

Concept: Kruskal's algorithm builds a minimum spanning tree by adding edges in strictly increasing order of weight, skipping any edge that would close a cycle among vertices already connected. When completing the tree requires choosing exactly one further edge from among several tied, equally valid edges — each joining two vertices that are still in different components — every valid choice at that final step gives a distinct minimum-weight spanning tree, so the number of MSTs equals the number of such valid choices.

Application:

1. The graph has 9 vertices, so any spanning tree needs exactly 8 edges.

2. There are 7 edges of weight 0.1 and 5 edges of weight 0.9. Kruskal's algorithm always adds the lowest available weight first, so all 7 weight-0.1 edges are considered before any weight-0.9 edge.

3. Adding all 7 weight-0.1 edges creates no cycle: they merge the graph into exactly two components — the three vertices of the top row (using 2 of the weight-0.1 edges), and the remaining six vertices of the lower block (using the other 5 weight-0.1 edges). That accounts for all 7 edges and all 9 vertices, split into a 3-vertex component and a 6-vertex component.

4. One more edge is now needed to join the two components into a single tree (7 + 1 = 8 edges total). Among the 5 remaining weight-0.9 edges, exactly 3 run from a vertex in the top-row component to a vertex in the lower block; the other 2 run between two vertices that are already inside the lower block, so adding either of those would close a cycle instead of extending the tree.

5. Since all 3 of those cross-component edges share the same weight (0.9) and each individually completes a valid, cycle-free spanning tree, each gives rise to a distinct minimum-weight spanning tree.

Cross-check: Every such tree has the same total weight — 7 × 0.1 + 1 × 0.9 = 1.6 — confirming all three trees are genuinely tied for minimum weight. No lower-weight combination is possible, since every weight-0.1 edge is forced: skipping any one of them would leave a vertex reachable only through a higher-weight edge.

Answer: The graph has exactly 3 distinct minimum-weight spanning trees.