Let $G = (V, E)$ be *any* connected undirected edge-weighted graph. The weights of the edges in $E$ are positive and distinct. Consider the following statements: (I) Minimum Spanning Tree of $G$ is always unique. (II) Shortest path between any two vertices of $G$ is always unique. Which of the above statements is/are necessarily true?

• Statement (I) is TRUE: When all edge weights in a connected, undirected graph are distinct, the Minimum Spanning Tree (MST) is guaranteed to be unique. This is a fundamental property of MSTs (e.g., via the Cut or Cycle property).

• Statement (II) is FALSE: Even if all individual edge weights are distinct and positive, two different paths between a pair of vertices can have the exact same total path weight. For example, consider a graph with 4 vertices $u, v, w, z$ where edge weights are $e(u, v) = 1$, $e(v, z) = 4$ (path length = $5$) and $e(u, w) = 2$, $e(w, z) = 3$ (path length = $5$). All edge weights are distinct ($\{1, 2, 3, 4\}$), but the shortest path between $u$ and $z$ is not unique.

Therefore, only statement (I) is necessarily true, making (A) the correct option.