I need to show that, given a connected graph with distinct weights for every edge, every spanning tree (except from the minimum spanning tree) has an adjacent spanning tree with a smaller weight. w(T') < w (T), where T' is adjacent to the spanning tree T.
I'm stuck at proving that every single ST that is adjacent to the MST has an adjacent spanning tree (the MST, indeed). How can I show this with whatever non-MST adjacent spanning tree?
The proof of the Kruskals algorithm answers your question. http://people.qc.cuny.edu/faculty/christopher.hanusa/courses/634sp12/Documents/KruskalProof.pdf
Let T be your MST and N be your non spanning tree. Since N!=T, find an edge e in T of minimum weight that is not in N.
N+e has a cycle where e is part of the cycle and there has to be a e' on the cycle such that wt(e)
So construct N'=N-e'+e. Obviously wt(N1)