Consider the following undirected graph with edge weights as shown: The number of minimum-weight spanning trees of the graph is _________.

The graph contains $9$ vertices arranged in a $3 \times 3$ grid. A spanning tree must contain $9 - 1 = 8$ edges.

1. Edges of weight $0.1$:

   • There are $7$ edges with weight $0.1$.

   • These $7$ edges do not contain any cycle and connect the $9$ vertices into $2$ disjoint connected components:

     • Component 1: $\{1, 2, 3\}$ (the top row)

     • Component 2: $\{4, 5, 6, 7, 8, 9\}$ (the middle and bottom rows)

   • By Kruskal's algorithm, all $7$ edges of weight $0.1$ must be included in any minimum spanning tree.

2. Edges of weight $0.9$:

   • To connect the two components into a single spanning tree, exactly $1$ edge of weight $0.9$ that crosses between Component 1 and Component 2 must be chosen.

   • The available crossing edges between the top row and middle row are the three vertical edges: $(1, 4)$, $(2, 5)$, and $(3, 6)$.

   • Any other edge of weight $0.9$ connects vertices within Component 2 and would create a cycle.

3. Therefore, there are 3 possible choices to complete the MST, resulting in $3$ distinct minimum-weight spanning trees of total weight: