TUYMAADA 2023 S2

In a graph with $n$ vertices every two vertices are connected by a unique path. For each two vertices $u$ and $v$ , let $d(u, v)$ denote the distance between $u$ and $v$ , i.e. the number of edges in the path connecting these two vertices, and $\deg(u)$ denote the degree of a vertex $u$ . Let $W$ be the sum of pairwise distances between the vertices, and $D$ the sum of weighted pairwise distances: $\sum_{\{u, v\}}(\deg(u)+\deg(v))d(u, v)$ . Prove that $D=4W-n(n-1)$ .