Peripheral Wiener index of trees and related questions

Distance-based topological indices, as a class of graph invariants, have received much attention. In particular, the Wiener index (sum of distances between all pairs of vertices) and terminal Wiener index (sum of distances between all pairs of leaves) are two of the most well known such indices in Chemical Graph Theory. Inspired by these concepts, the peripheral Wiener index is recently introduced as the sum of distances between all pairs of peripheral vertices (vertices with maximum eccentricity). In this note we consider a number of interesting problems related to this new distance-based index and propose some potential topics for further study.