Minimum Spanning Tree

Let G be a connected undirected weighted graph. Consider the following two statements.

S1: There exists a minimum weight edge in G which is present in every minimum spanning tree of G.
S2: If every edge in G has distinct weight, then G has a unique minimum spanning tree.

Which one of the following options is correct?

Consider the following undirected graph with edge weights as shown:

The number of minimum-weight spanning trees of the graph is __________

Consider a graph G=(V,E), where V=\{v_1,v_2,...,v_{100}\}, E=\{(v_i,v_j)|1\leq i \lt j\leq 100\}, and weight of the edge (v_i,v_j)\; is \; |i-j|. The weight of minimum spanning tree of G is _________

Let G=(V,E) be a weighted undirected graph and let T be a Minimum Spanning Tree (MST) of G maintained using adjacency lists. Suppose a new weighed edge (u,v)\in V \times V is added to G. The worst case time complexity of determining if T is still an MST of the resultant graph is

Let G be any connection, weighted, undirected graph:

I. G has a unique minimum spanning tree if no two edges of G have the same weight.
II. G has a unique minimum spanning tree if, for every cut of G, there is a unique minimum weight edge crossing the cut.

Which of the above two statements is/are TRUE?

Consider the following undirected graph G:

Choose a value for x that will maximize the number of minimum weight spanning trees (MWSTs) of G. The number of MWSTs of G for this value of x is ______.

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?

G = (V,E) is an undirected simple graph in which each edge has a distinct weight,and e is a particular edgeof G. Which of the following statements about the minimum spanning trees (MSTs) of G is/are TRUE?
I. If e is the lightest edge of some cycle in G, then every MST of G includes e
II. If e is the heaviest edge of some cycle in G, then every MST of G excludes e

Let G be a complete undirected graph on 4 vertices, having 6 edges with weights being 1,2,3,4,5, and 6. The maximum possible weight that a minimum weight spanning tree of G can haveis .

Let G be a weighted connected undirected graph with distinct positive edge weights. If every edge weight is increasedby the same value ,then which of the following statements is/are TRUE?
P: Minimum spanning tree of G does not change
Q: Shortest path between any pair of vertices does not change