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.

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.