https://file.aiquickdraw.com/mathbot/file/1729638304242xi8m84mv.jpg

The problem asks to apply the **Nearest Neighbor Algorithm** to find an approximate **Minimum Hamiltonian Circuit** for the given graph, starting at vertex A.

### Steps to Apply the Nearest Neighbor Algorithm:
1. **Start at vertex A**.
2. At each step, move to the closest vertex (one with the smallest edge weight that has not been visited yet).
3. Repeat the process until all vertices are visited.
4. Return to the starting vertex A to complete the circuit.

### Applying the Nearest Neighbor Algorithm:

- **Start at A**:
    - From A, the closest neighbor is C (edge weight = 1).
    - Path: **A → C**

- **At C**:
    - From C, the closest unvisited neighbor is B (edge weight = 2).
    - Path: **A → C → B**

- **At B**:
    - From B, the closest unvisited neighbor is D (edge weight = 6).
    - Path: **A → C → B → D**

- **At D**:
    - From D, the closest unvisited neighbor is E (edge weight = 4).
    - Path: **A → C → B → D → E**

- **At E**:
    - The only unvisited vertex left is F (edge weight = 8).
    - Path: **A → C → B → D → E → F**

- **At F**:
    - Now return to A (edge weight = 7).
    - Path: **A → C → B → D → E → F → A**

### Final Approximate Minimum Hamiltonian Circuit:
The path following the Nearest Neighbor Algorithm is:
\[ **A → C → B → D → E → F → A** \]

### Total weight of the path:
1 (A to C) + 2 (C to B) + 6 (B to D) + 4 (D to E) + 8 (E to F) + 7 (F to A) = **28**.

Let me know if you need more details or clarifications!

### Related Questions:
1. How does the Nearest Neighbor Algorithm compare to other algorithms for the Traveling Salesman Problem?
2. Can the Nearest Neighbor Algorithm guarantee the optimal Hamiltonian circuit?
3. What are some limitations of the Nearest Neighbor Algorithm?
4. How would the solution change if starting from a different vertex?
5. What is the significance of edge weights in the context of real-world applications?

### Tip:
When using the Nearest Neighbor Algorithm, the order of visiting vertices can vary significantly depending on the starting point, potentially leading to different solutions with varying total weights.

Apply the Nearest Neighbor Algorithm to find an approximate Minimum Hamiltonian Circuit for the graph above. Start at vertex A. Give your answer as a list of vertices, starting and ending at vertex A.

This problem involves using the Nearest Neighbor Algorithm to approximate the Minimum Hamiltonian Circuit, starting at vertex A. The graph includes 6 vertices with given edge weights. The total path weight is 28. The problem helps illustrate graph theory concepts like the Traveling Salesman Problem and optimization strategies.

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