Inverted distance and inverted Wiener index (Q2802258)

The inverted distance between any two different vertices \(u\) and \(v\) of a simple connected graph \(G\) is defined as \(i(u,v)=D-d(u,v)+1\), where \(D\) denotes the diameter of \(G\) and \(d(u,v)\) the distance between \(u\) and \(v\). The inverted Wiener index of a simple connected graph \(G\) is defined as \(\mathrm{IW}(G)=\sum_{u\neq v}i(u,v)\), the sum is taken over all unordered pairs of vertices of \(G\). The authors characterize the maximum trees with respect to the inverted Wiener index. They prove the inequality \(\mathrm{IW}(S_n)\leq \mathrm{IW}(T)\leq \mathrm{IW}(P_n)\), where \(S_n\) is the star graph \(K_{1,n-1}\), \(T\) is a tree and \(P_n\) is a path on \(n\geq 3\) vertices.