Prove that if G is a connected, weighted graph and e is an edge of G that (1) has greater weight than any other edge of G and (2) is in a circuit of G, then there is no minimum spanning tree T for G such that e is in T.

To prove:

When G is a connected weighted graph and e is an edge of G which is in a circuit of G and that has larger weight than any other edge of graph G, then there exists no minimum spanning tree T for G such that the edge e is in T.

Given information

G: Weighted graph

T1,T2: Distinct minimum spanning trees.

Concept used:

G' contains a nontrivial circuit.

Proof:

Suppose that there is a minimal spanning tree T for G such that e is in T.

Note that e is an edge of G and has larger weight than any other edge.

It is in a circuit of G.

Let P be another tree which has the same edges as in T except e