The problem involves finding the shortest path from vertex A to vertex B in a weighted graph. Each edge in the graph has a weight associated with it, representing the distance between two vertices. I will use Dijkstra's algorithm to find the shortest path from vertex A to vertex B.

Steps:

1. Identify all vertices: $A$, $B$, $C$, $D$, and $E$.
2. List the edges and their weights:
   • $A \to B: 9$
   • $A \to C: 6$
   • $A \to D: 20$
   • $A \to E: 17$
   • $B \to C: 7$
   • $B \to E: 27$
   • $C \to D: 24$
   • $D \to E: 30$

Next, I will calculate the shortest path using Dijkstra's algorithm.

Let me compute this path for you.The shortest path from vertex $A$ to vertex $B$ is through the direct edge between them, with a total distance of 9. The path is simply $A \to B$.

Would you like more details or have any further questions?

Here are 5 related questions for further exploration:

1. How does Dijkstra's algorithm work in weighted graphs?
2. Can Dijkstra’s algorithm handle negative weights?
3. What is the time complexity of Dijkstra's algorithm for a graph with $n$ vertices?
4. How would the shortest path change if the weights on edges were dynamic?
5. How does Dijkstra’s algorithm differ from the Bellman-Ford algorithm?

Tip: Dijkstra's algorithm is efficient when all edge weights are positive, but for graphs with negative weights, consider using Bellman-Ford's algorithm instead.