https://file.aiquickdraw.com/mathbot/file/17331065036323w6qbn4p.jpg

Let’s solve the questions step-by-step based on the given weighted graph and instructions:

### **4. Using Prim's algorithm (starting at vertex \(e\)):**
Prim's algorithm starts at a specific vertex (here, \(e\)) and grows the minimum spanning tree (MST) by adding the smallest-weight edge connecting the tree to a vertex not yet in the tree.

#### Steps:
1. Start at vertex \(e\). The edges connected to \(e\) are:
   - \(e \to f\) (weight = 5)
   - \(e \to g\) (weight = 6)
   - \(e \to h\) (weight = 7)
   - \(e \to d\) (weight = 8)

2. Select the smallest edge: \(e \to f\) (weight = 5). Add \(f\) to the MST.

3. Add the smallest edge connecting \(f\) or \(e\) to a new vertex:
   - \(f \to g\) (weight = 4)
   - \(f \to h\) (weight = 6)
   - \(f \to i\) (weight = 4)
   - \(e \to g\) (weight = 6)
   - \(e \to h\) (weight = 7)
   - \(e \to d\) (weight = 8)

   Select \(f \to g\) (weight = 4). Add \(g\) to the MST.

4. Repeat for the remaining vertices, adding the smallest edge without forming a cycle:
   - Add \(f \to i\) (weight = 4).
   - Add \(f \to h\) (weight = 6).
   - Add \(e \to d\) (weight = 8).

#### MST Edges:
\(\{(e, f), (f, g), (f, i), (f, h), (e, d)\}\)

---

### **5. Using Kruskal's algorithm:**
Kruskal's algorithm sorts all edges by weight and selects edges in ascending order, avoiding cycles, until the MST is formed.

#### Steps:
1. Sort edges:
   - \(f \to g\) (4)
   - \(f \to i\) (4)
   - \(a \to b\) (3)
   - \(a \to d\) (3)
   - \(e \to f\) (5)
   - \(f \to h\) (6)
   - \(e \to g\) (6)
   - \(e \to h\) (7)
   - \(e \to d\) (8)

2. Select edges in ascending order:
   - \(f \to g\) (4)
   - \(f \to i\) (4)
   - \(a \to b\) (3)
   - \(a \to d\) (3)
   - \(e \to f\) (5)
   - \(f \to h\) (6)

#### MST Edges:
\(\{(a, b), (a, d), (e, f), (f, g), (f, h), (f, i)\}\)

---

### **6. Total weight of MST:**
The sum of weights for the edges in the MST:
\[
4 + 4 + 3 + 3 + 5 + 6 = 25
\]

---

### **7. Using Prim's algorithm (starting at \(d\)):**
Follow the same process as in Question 4 but start at \(d\).

#### Steps:
1. Start at \(d\). The edges connected to \(d\) are:
   - \(d \to a\) (weight = 3)
   - \(d \to e\) (weight = 8)

   Select \(d \to a\) (weight = 3). Add \(a\) to the MST.

2. Add the smallest edge:
   - \(a \to b\) (weight = 3)
   - \(a \to e\) (weight = 6)

   Select \(a \to b\) (weight = 3).

3. Continue until all vertices are included.

#### MST Edges:
Similar edges will result, since the MST structure is the same regardless of the starting vertex.

\(\{(a, b), (a, d), (e, f), (f, g), (f, h), (f, i)\}\)

---

### Do you want detailed explanations of any step? Here are five questions you might explore further:

1. How do Prim's and Kruskal's algorithms differ in approach and use cases?
2. What happens if a different starting vertex is chosen for Prim's algorithm?
3. Could there be multiple MSTs for the same graph? Why or why not?
4. How would the algorithm handle negative edge weights in the graph?
5. What is the computational complexity of Prim’s and Kruskal’s algorithms?

**Tip:** Always visualize the graph when applying MST algorithms to understand the tree's growth!

Using Prim's and Kruskal's algorithms to find the Minimum Spanning Tree (MST) of a weighted graph.

Explore step-by-step solutions to finding the Minimum Spanning Tree (MST) of a weighted graph using Prim's and Kruskal's algorithms. Understand the core principles of graph theory, the greedy approach, and learn how to calculate MST weight efficiently.

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