https://file.aiquickdraw.com/mathbot/file/1729966370017vj6qn4uh.jpg

To find the weight of the minimum spanning tree (MST) for this graph, we can use either **Kruskal's** or **Prim's algorithm**. I will go through the steps here with Kruskal's algorithm:

### Steps in Kruskal's Algorithm
1. **Sort all edges** by their weights in ascending order.
2. **Add the smallest edge** to the MST as long as it doesn’t form a cycle.
3. **Repeat** until we have \( V - 1 \) edges in the MST (where \( V \) is the number of vertices).

### Sorted Edges
From the image, the edges with their weights are:
- \( AE = 10 \)
- \( BF = 10 \)
- \( AB = 15 \)
- \( BD = 15 \)
- \( CD = 25 \)
- \( CF = 25 \)
- \( AF = 30 \)
- \( BE = 30 \)
- \( ED = 35 \)

### Building the MST
Following Kruskal’s approach:

1. **Select \( AE \) and \( BF \)**, each with a weight of 10.
2. **Select \( AB \) and \( BD \)**, each with a weight of 15.
3. **Select \( CD \)** with a weight of 25.

These edges connect all vertices (A, B, C, D, E, F) without forming cycles, and the sum of these weights gives the minimum spanning tree.

### Calculation
Adding up the weights:
\[
10 + 10 + 15 + 15 + 25 = 75
\]

### Answer
The weight of the minimum spanning tree for the graph is **75**.

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Would you like a breakdown of any of these steps or more details on Kruskal’s algorithm?

### Related Questions
1. How would the solution change if we used Prim's algorithm instead?
2. What is the difference between Kruskal’s and Prim’s algorithms for finding an MST?
3. How would the solution be affected if an additional edge with a weight of 5 was added between two vertices?
4. Can we use this method on a directed graph? Why or why not?
5. How does the concept of a minimum spanning tree apply in real-world applications?

### Tip
When applying Kruskal's algorithm, always start by sorting the edges by weight. This step helps in systematically building the MST with the minimum weight.

Find the weight of the minimum spanning tree for the graph.

Learn to find the weight of the minimum spanning tree for a graph using Kruskal’s algorithm. This problem involves concepts from graph theory and is suitable for high school students in Grades 11-12.

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