The task is to apply the nearest neighbor algorithm on the given graph starting at vertex A and ending at A, forming a Hamiltonian circuit. The algorithm works by choosing the nearest (smallest edge weight) unvisited neighbor at each step.

Let's go step-by-step:

1. Start at vertex A.

   • The edges connected to A are:
     • A to B: weight 6
     • A to C: weight 9
     • A to F: weight 13
     • A to E: weight 10
   • The smallest edge is A → B (weight 6).

2. Move to vertex B.

   • The edges connected to B are:
     • B to A: weight 6 (already visited)
     • B to C: weight 3
     • B to D: weight 5
     • B to F: weight 8
   • The smallest edge is B → C (weight 3).

3. Move to vertex C.

   • The edges connected to C are:
     • C to A: weight 9
     • C to B: weight 3 (already visited)
     • C to D: weight 14
     • C to E: weight 12
   • The smallest edge is C → E (weight 12).

4. Move to vertex E.

   • The edges connected to E are:
     • E to A: weight 10
     • E to C: weight 12 (already visited)
     • E to D: weight 7
     • E to F: weight 15
   • The smallest edge is E → D (weight 7).

5. Move to vertex D.

   • The edges connected to D are:
     • D to A: weight 9
     • D to B: weight 5 (already visited)
     • D to C: weight 14 (already visited)
     • D to E: weight 7 (already visited)
   • The smallest edge is D → A (weight 9).

Now the circuit is complete, and the path is A → B → C → E → D → A.

Would you like any further details or clarification?

Here are 5 related questions:

1. What are the other algorithms used for solving Hamiltonian circuits?
2. How does the nearest neighbor algorithm compare to the brute force method?
3. Can the nearest neighbor algorithm always guarantee the shortest possible path?
4. What is the time complexity of the nearest neighbor algorithm for Hamiltonian circuits?
5. How would the nearest neighbor algorithm behave if the graph was directed?

Tip: The nearest neighbor algorithm is a greedy algorithm and does not always yield the optimal solution but provides a quick approximation.