Extremal properties of reciprocal complementary Wiener number of trees

The reciprocal complementary Wiener (RCW) number of a connected graph GG is defined in mathematical chemistry as the sum of the weights 1d+1−dG(u,v) of all unordered pairs of distinct vertices, where dd is the diameter and dG(u,v)dG(u,v) is the distance between vertices uu and vv in GG. Among others, we characterize the trees of fixed number of vertices and matching number with the smallest RCW number, and the trees that are not caterpillars on n≥7n≥7 vertices with the smallest, the second-smallest and the third-smallest RCW numbers.