# GATE | GATE CS Mock 2018 | Question 37

• Last Updated : 03 Jan, 2018

Let G = (V, E) be any connected undirected edge-weighted graph. The weights of the edges in E are positive. Consider the following statements:

1. The path between a pair of vertices in a minimum spanning tree of an undirected graph is necessarily the shortest (minimum weight) path.
2. Minimum Spanning Tree of G is always unique and shortest path between a pair of vertices may not be unique.

Which of the above statements is/are necessarily true?
(A) (1) only
(B) (2) only
(C) both (1) and (2)
(D) neither (1) nor (2)

Explanation: Since, edge weights are not defined. If edge weights are distinct then MST is always unique but may not be shortest paths.

```(A)---1----(B)----2---(C)
\                    /
---------3----------
```

And, if edge weights are not distinct then both MST and shorted paths are not unique.

```(A)---2----(B)----2---(C)
\                    /
---------2----------
```

So, both statements are false.
Option (D) is correct.

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