00:01
We're asked to prove that there is a unique minimum spanning tree in a connected weighted graph if the weights of the edges are all distinct.
00:12
To do this, i'll use a proof by contradiction.
00:16
So let g be a connected weighted graph with n vertices.
00:39
Let t be a minimum spanning tree of g, which we know exists by either cruiscoles or prims algorithm.
01:03
And suppose that the weights of the edges in g are all different.
01:28
Now, for the contradiction, we're going to suppose u is a different minimum spanning tree for g, so it's different from t.
02:14
Then since they're both minimum spanning trees, it falls like the same weight, and so we're going to suppose that the total weight of t equals the total weight of u, which is equal to a.
02:49
Now since t and u are different trees, it follows that there exists an edge in t but not in u.
03:05
So find an edge e with vertices u and v in t but not in the graph u...