The problem asks to apply the "nearest neighbor" algorithm to the given weighted graph starting from vertex $A$ and return to $A$. The table provides the weights of edges between the vertices $A, B, C, D, E, F$.

Nearest Neighbor Algorithm:

1. Start from the initial vertex $A$.
2. From the current vertex, move to the nearest unvisited vertex (the one with the smallest edge weight).
3. Repeat this process until all vertices are visited.
4. Finally, return to the starting vertex $A$.

Step-by-Step Solution:

Start at $A$:

• The edge weights from $A$ to other vertices are:
  • $A \to B = 16$
  • $A \to C = 51$
  • $A \to D = 3$
  • $A \to E = 53$
  • $A \to F = 41$

The smallest weight is $3$ (to vertex $D$). Move to $D$.

From $D$:

• The edge weights from $D$ to other unvisited vertices are:
  • $D \to B = 1$
  • $D \to C = 30$
  • $D \to E = 34$
  • $D \to F = 50$

The smallest weight is $1$ (to vertex $B$). Move to $B$.

From $B$:

• The edge weights from $B$ to other unvisited vertices are:
  • $B \to C = 13$
  • $B \to E = 27$
  • $B \to F = 55$

The smallest weight is $13$ (to vertex $C$). Move to $C$.

From $C$:

• The edge weights from $C$ to other unvisited vertices are:
  • $C \to E = 46$
  • $C \to F = 11$

The smallest weight is $11$ (to vertex $F$). Move to $F$.

From $F$:

• The only unvisited vertex is $E$, so move to $E$ (edge weight = 48).

Return to $A$:

• Finally, return to $A$ from $E$ (edge weight = 53).

Final Path:

The nearest neighbor tour is: $A \to D \to B \to C \to F \to E \to A$

Would you like further details on how this algorithm works, or how to calculate total path weight? Here are some further questions that might help:

1. How does the nearest neighbor algorithm compare to other pathfinding algorithms?
2. Can this method guarantee the shortest possible path for every case?
3. How would the solution change if we started at a different vertex?
4. Could you explain how the table of edge weights was used to make decisions?
5. What is the total distance or weight for this nearest neighbor path?

Tip: The nearest neighbor algorithm is a heuristic, so while it's fast and simple, it doesn't always find the optimal solution for larger or more complex graphs!