Does the minimum spanning tree (MST) of a weighted graph always give the shortest distance between any two specified nodes in that graph?

Introduction / Context:
Graph algorithms play a major role in computer science and networking. Two important problems are finding shortest paths between nodes and finding a minimum spanning tree. Although both deal with edge weights, they solve different optimization problems. This question tests whether you understand that the minimum spanning tree does not necessarily provide shortest path distances between arbitrary pairs of nodes.

Given Data / Assumptions:

• We have a connected weighted undirected graph.
• A minimum spanning tree is a subset of edges that connects all vertices with minimum total weight and contains no cycles.
• Shortest path between two nodes is defined as the path with minimum sum of edge weights between them.
• We assume non negative edge weights but do not impose special restrictions beyond that.

Concept / Approach:
The minimum spanning tree problem and the shortest path problem optimize different objectives. An MST minimizes the sum of the weights of all edges used to connect all vertices. It does not consider any single pair of vertices in isolation. In contrast, a shortest path algorithm such as Dijkstra's algorithm focuses on finding the least cost path between a specific source and destination. Because these objectives differ, an edge that is important for a particular shortest path might be excluded from the MST if it does not contribute to the minimum total tree cost.

Step-by-Step Solution:
Step 1: Recall the definition of a minimum spanning tree as a tree connecting all vertices with minimum total weight.Step 2: Recall that the shortest path between two nodes is the path with the minimum sum of edge weights between them.Step 3: Consider that minimizing total tree weight and minimizing each pairwise shortest path are different optimization goals.Step 4: Recognize that the MST may omit edges that are part of shortest paths but not needed for the minimum total weight tree.Step 5: Conclude that the MST does not always give shortest distances between arbitrary node pairs.

Verification / Alternative check:
You can construct a counterexample to verify this. Imagine a graph where a direct edge between two vertices provides the shortest path between them but this edge has a relatively high weight compared to other edges that can still connect all vertices. The MST algorithm might exclude this heavy edge to minimize total weight, choosing a longer path between those two vertices in the tree. In such a case, the path between the two vertices in the MST is not the original shortest path in the full graph, confirming that MST and shortest path trees are different.

Why Other Options Are Wrong:
Option A claims that the MST always gives shortest distances between any two nodes, which is false because of the differing optimization goals. Option C says this holds only for complete graphs with equal edge weights, but in that trivial case many trees are minimum, and shortest paths are all equal, so the statement is not meaningful and does not capture the general relationship. Option D talks about directed graphs, but classical MST definitions and algorithms apply to undirected graphs, and even with directed analogs the MST concept still does not guarantee shortest paths.

Common Pitfalls:
A frequent mistake is to conflate MST algorithms like Prim or Kruskal with shortest path algorithms such as Dijkstra or Bellman Ford. Students sometimes assume that minimizing something involving weights must also minimize all related metrics, which is not true here. Another pitfall is to think that because the MST is a tree connecting all vertices, it must be optimal for all path queries, which is again incorrect. Keeping the objectives of each algorithm clearly in mind helps avoid these misunderstandings.

Final Answer:
The minimum spanning tree does not guarantee shortest distances between any two specified nodes; it minimizes total tree weight only. The correct option is B, No, the MST minimizes total edge weight, not pairwise shortest paths.