If G is a connected, weighted graph and no two edges of G have the same weight, does there exist a unique minimum spanning tree for G? Use the result of exercise 19 help justify your answer.

To check:

If G is a connected, weighted graph and no two edges of G have the same weight, then whether there exist a unique minimum spanning tree for G.

Given information

G: Weighted graph

T1,T2: Distinct minimum spanning trees.

Concept used:

G' contains a nontrivial circuit.

Calculation:

At least one other edge f of this circuit is not in T1 and e is the edge of least weight that is in one of the trees and not the other. Remove f from G' to obtain a tree T3. Then, w(T3)<w(T2) because T3 is same as T2