S1: “There exists a minimum weight edge in G”
There can be two interpretations of this statement-
1) There exists a specific minimum weight edge which must be present in all minimum spanning trees of the graph.
2) There exists atleast one minimum weight edge which is present in every minimum spanning trees of the graph.
In either case S1 will be false statement. Let me prove this using two examples.
For case 1
Here edge AB is not present in two of the possible minimum spanning trees.
For case 2
Here no edge is common between the two possible minimum spanning trees of the graph. That is there does not exist atleast 1 edge which is present in the two possible spanning trees.
So we can clearly state that Statement 1 is False.
S2: “If every edge in G has distinct weight, then G has a unique minimum spanning tree.”
This statement is obvious since if edge weights are distinct there will always be a distinct edge set and only 1 minimum spanning tree possible. Which is True
Correct Answer C