In a given weighted graph shown below, what is the value of the expression \( (p + d)^2 \), where: [i.] Alphabets A, B, C, D, E, and F denote the nodes [ii.] Numbers 1 to 6 denote the weights between two nodes [iii.] \( d \) = shortest distance between node A and node E [iv.] \( p \) = number of paths with distance \( d \)
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In weighted graphs, identify the shortest path using path enumeration or Dijkstra’s algorithm. To compute expressions involving path count, list all distinct paths with the same minimal distance.
We are given a weighted undirected graph and asked to evaluate the expression \( (p + d)^2 \), where:
- \( d \) = shortest distance from node A to node E
- \( p \) = number of paths from A to E with that distance
Let us evaluate all possible paths from A to E:
1. \( A \rightarrow B \rightarrow E \): \( 3 + 3 = 6 \)
2. \( A \rightarrow D \rightarrow E \): \( 1 + 5 = 6 \)
3. \( A \rightarrow D \rightarrow F \rightarrow E \): \( 1 + 2 + 3 = 6 \)
Other paths like \( A \rightarrow B \rightarrow D \rightarrow E \) or those involving node C are longer than 6.
Hence, the shortest distance is:
\[
d = 6
\]
and the number of such shortest paths is:
\[
p = 3
\]
Therefore, the value of the expression is:
\[
(p + d)^2 = (3 + 6)^2 = 9^2 = \boxed{81}
\]
