A proof that if f is the heaviest edge in weight from all the other edges in the circle which it is a part of, then f will not participate in any MST

Question

I cant proof that if f is the heaviest edge in weight from all the other edges in the circle which it is a part of, then f will not participate in any Minimum Spanning Tree. please help.

Answer 1

Lets assume that it did participate in an MST $T$.

Now, removing $f$ will split $T$ into two connected components - $T_1,T_2$. In particular, both vertices of $f$ need to be in different connected components. Lets call the vertices $v_1$ and $v_k$, and w.l.o.g. assume $v_1\in T_1$ and $v_k\in T_2$.

Lets name all vertices in the cycle by $v_1,v_2,\dots,v_k$ such that there is an edge between $v_i$ and $v_{i+1}$ for all $i$ (and of course, $v_1,v_k$ are the vertices of $f$).

Since $v_1\in T_1$ and $v_k\in T_2$, and also $v_i\in T_1\lor v_i\in T_2$ then there must be some index $i_0<k$ with $v_{i_0}\in T_1$ and $v_{i_0+1}\in T_2$. We know that $(v_{i_0} , v_{i_0 + 1})$ is an edge in $G$, and its weight is lower than that of $f$ - and in addition it will connect the two components! Therefore, by replacing $f$ with this edge we get another spanning tree with a lower weight, which is a contradiction to the assumption we started with an MST!