How Does Unique Edge Weighting Guarantee a Singular Minimum Spanning Tree?

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SUMMARY

The discussion confirms that a weighted graph with unique edge weights guarantees a singular minimum spanning tree (MST). It establishes that using Kruskal's Algorithm, which operates on the principle of selecting edges in increasing order of weight, results in a unique sequence of (n-1) edges. The proof involves contradiction, demonstrating that if two MSTs exist, at least one edge must differ, leading to a contradiction. Thus, the uniqueness of edge weights directly correlates with the uniqueness of the MST.

PREREQUISITES
  • Understanding of weighted graphs
  • Familiarity with Kruskal's Algorithm
  • Knowledge of minimum spanning trees (MST)
  • Basic principles of proof by contradiction
NEXT STEPS
  • Study the implementation of Kruskal's Algorithm in Python
  • Explore the properties of unique edge weights in graph theory
  • Learn about Prim's Algorithm and its comparison with Kruskal's
  • Investigate other proofs of uniqueness for minimum spanning trees
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Students of computer science, algorithm enthusiasts, and professionals in graph theory seeking to deepen their understanding of minimum spanning trees and their properties.

find_the_fun
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Prove that if a weighted graph has unique weights on each edge that there is only 1 minimum spanning tree. Do I need to use an algorithm such as Prim's to show this or do I use properties of weighted graphs?
 
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Try contradiction. Assume you have two MST's of equal weight. Consider what happens with an edge that's in one MST but not the other (there must be at least one such edge if the MST's are not the same).
 
Kruskal's Algorithm can be used here as follows:

If edge costs are all distinct then running Kruskal's Algorithm on this graph G(say) creates only a unique sequence of (n-1) edges with edge costs in increasing order. Hence G has a unique minimum spanning tree.
 

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