The number of distinct minimum-weight spanning trees of the following graph is…

Key idea: use Kruskal's approach: take all smallest-weight edges that do not create a cycle, then count choices when there are ties.

• List edges by weight. Edges of weight 1 are: a–b, a–f, c–d, e–d. These four do not form any cycle, so any minimum spanning tree (MST) must include all four.

• After including the weight‑1 edges, the graph breaks into three components: {a,b,f}, {c,d,e}, and {g}.

• An MST on 7 vertices needs 6 edges total; we already have 4, so we must add exactly 2 more edges to connect the three components.

• All edges of weight 2 connect the center g to vertices in the other two components. The edges from g to the left component are a–g, b–g, f–g (3 choices). The edges from g to the right component are g–c, g–d, g–e (3 choices).

• To connect the three components without creating a cycle we must pick one edge from the left set and one from the right set. That gives 3 × 3 = 9 distinct ways.

Conclusion: The number of distinct minimum-weight spanning trees is 9.