Question

In the given weighted graph, the weights represent distance between the corresponding vertices. The shortest distance between vertex A and vertex E is \( d \).
The number of paths with distance \( d \) is \( p \).
The value of \( d/p \) is ______.

• A. 5
• B. 3
• C. 2
• D. 8

Hint:

List every possible route from A to E along the given edges, add up the weights, and find the minimum total distance and how many routes achieve it.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given a network with six nodes, A to F, connected by edges of given weights (distances).
We must find the shortest total distance \( d \) from A to E, count how many different paths achieve that exact shortest distance, call this count \( p \), and then compute \( d/p \).

Step 2: Key Formula or Approach:
From the figure, the edges and their weights are: \( A\text{-}B = 2 \), \( A\text{-}C = 1 \), \( B\text{-}C = 1 \), \( B\text{-}D = 5 \), \( B\text{-}E = 4 \), \( C\text{-}F = 2 \), \( D\text{-}E = 3 \), \( F\text{-}E = 3 \).
Since the network is small, the simplest approach is to list every reasonable simple path from A to E and add up its edge weights.

Step 3: Detailed Explanation:
Path \( A \to B \to E \): distance \( = 2 + 4 = 6 \).
Path \( A \to C \to B \to E \): distance \( = 1 + 1 + 4 = 6 \).
Path \( A \to C \to F \to E \): distance \( = 1 + 2 + 3 = 6 \).
Path \( A \to B \to D \to E \): distance \( = 2 + 5 + 3 = 10 \).
Path \( A \to C \to B \to D \to E \): distance \( = 1 + 1 + 5 + 3 = 10 \).
The smallest of these totals is \( 6 \), reached by exactly three different paths.
\[ d = 6, \quad p = 3 \]

Final Answer:
Dividing the shortest distance by the number of shortest paths gives the answer.
\[ \frac{d}{p} = \frac{6}{3} = 2 \]
\[ \boxed{2} \]