The number of distinct minimum spanning trees for the weighted graph below is…

Answer: 6 distinct minimum spanning trees.

Reasoning (use Kruskal's idea): include all smallest-weight edges unless they create a cycle, and handle ties carefully.

• All edges labelled 1 in the drawing must be in every minimum spanning tree because they are strictly lighter than the alternative edges that would connect the same parts and they never create a cycle when taken together.

• There are two independent tie situations among the weight-2 edges where you have a choice:

  • Top-left triangle: all three edges of that small triangle have weight 2. To span its three vertices you must pick any two of those three edges. That gives 3 possible choices for this triangle (one can omit any one of the three edges).

  • Rightmost small triangle: its edges have weights 1, 2, and 2. The weight-1 edge must be included, and then you must choose one of the two weight-2 edges to complete that triangle without forming a cycle. That gives 2 possible choices for this triangle.

• All other edges (the connecting horizontals and bridges shown with weight 2) are forced by connectivity once the 1-edges and the chosen edges from the tied triangles are fixed.

Because the two tie decisions are independent, multiply the counts: 3 choices from the top-left triangle times 2 choices from the rightmost triangle = 6 distinct minimum spanning trees in total.