Minimum Spanning Tree over Vertices Proof
It doesn't matter which minimum spanning tree algorithm you use. All you need to notice is that for a tree $T$,
$$
\begin{align*}
\sum_{(i,j) \in T} m(e_{ij}) &amp;= \sum_{(i,j) \in T} w(v_i) + w(v_j) \\ &amp;=
\sum_i \sum_{(i,j) \in T} w(v_i) + \sum_j \sum_{(i,j) \in T} w(v_j) \\ &amp;=
2 \sum_i d_T(i) w(v_i).
\end{align*}
$$
You take it from here.