Unique Minimum Spanning Tree Condition
In a weighted, connected graph, explain why a unique Minimum Spanning Tree (MST) exists if all the edge weights are distinct. Your answer should reference the cut property in your explanation.
Uniqueness of Minimum Spanning Trees
Explain, using the cut property, why a minimum spanning tree (MST) is unique if all edge weights in the graph are distinct.
Proof of Minimal Spanning Tree Minimality
Given a weighted, connected graph, prove that a Minimum Spanning Tree (MST) is indeed minimal. In your proof, clearly define a ‘cut’ and explain the role of the cut property. Demonstrate, using proof by contradiction, that no other spanning tree connecting all vertices can have a lower total weight.