Let $G$ be a connected undirected weighted graph. Consider the following two statements. $S_1$ : There exists a minimum weight edge in $G$ which is present in every minimum spanning tree of $G$. $S_2$ : If every edge in $G$ has distinct weight, then $G$ has a unique minimum spanning tree. Which one of the following options is correct?

Let us analyze both statements:

1. Statement $S_1$: *"There exists a minimum weight edge in $G$ which is present in every minimum spanning tree of $G$."*

   • Consider a cycle graph $C_3$ with 3 vertices $u, v, w$ and edges $(u,v), (v,w), (w,u)$, all having the same minimum weight $1$.

   • Any spanning tree of this graph consists of any 2 of the 3 edges, yielding $\binom{3}{2} = 3$ distinct Minimum Spanning Trees (MSTs): $T_1 = \{(u,v), (v,w)\}$, $T_2 = \{(v,w), (w,u)\}$, and $T_3 = \{(u,v), (w,u)\}$.

   • No single edge is present in all three MSTs. Hence, $S_1$ is false [cite: 1].

2. Statement $S_2$: *"If every edge in $G$ has distinct weight, then $G$ has a unique minimum spanning tree."*

   • By standard graph theory (the Cut and Cycle properties), if all edge weights in a connected undirected graph are strictly distinct, the minimum spanning tree is unique [cite: 1]. Hence, $S_2$ is true [cite: 1].

Therefore, $S_1$ is false and $S_2$ is true, making (C) the correct option.