A Note on Plesnik’S Long-Open Standing Problem

The classical papers of plesnik (1975) and (1984) are two famous manuscripts in metric graph theoryin which some pretty and long-standing problems were posed. plesnik determined some sharp lower andupper bounds for an average distance of graph and digraph with given order and diameter. in thesearticles, he addressed some problems on the average distance of graphs which remains unsolved for a longtime. one of the most well-known of these problems is as follows:problem [plesnik (1975)]: what is the maximum average distance among all graphs of given orderand diameter?the main contribution of this paper is to investigate the long-standing open problem posed by plesnikin 1975. we prove that if t is a tree with a maximum average distance of given order and diameter thent is a centered tree such thatthe branches of the center of t are almost isomorphic (differ by at most onependant vertex at the last level) and the degrees of vertices except the center are at most three. moreover,the leaves are at the last level that is, the distance r from the center. furthermore, we illustrate a class oftrees with a maximum average distance of given order and diameter such that the obvious bound µ(g) ≤ dis best possible.