Question

The number of distinct minimum-weight spanning trees of the following graph is ............ \begin{center} \includegraphics[width=0.5\textwidth]{59.png} \end{center}

Hint:

When determining the number of distinct MSTs, consider the flexibility in selecting edges of the same weight without violating the MST property.

Answer 1

Step 1: Analyze the graph and find the minimum weight. The graph has 7 edges with weights: \(2, 2, 2, 3, 3, 3, 1\). To construct a minimum spanning tree (MST), the sum of edge weights must be minimized. 

Step 2: Apply Kruskal's algorithm. Using Kruskal's algorithm: Select the edge with weight \(1\) (unique choice). Choose three edges of weight \(2\). These edges form a cycle, allowing multiple choices. Choose one edge of weight \(3\) to complete the MST. 

Step 3: Count the distinct combinations. There are \(\binom{3}{2} = 3\) ways to choose two edges of weight \(2\). For each choice, there are \(3\) ways to select one edge of weight \(3\). The total number of distinct MSTs is: \[ 3 \times 3 = 9. \] 


Final Answer: \[ \boxed{9} \]