Abstract
The Wiener index, defined as the sum of distances between all pairs of vertices, is the oldest and most studied distance-based topological index that has found many applications in chemistry as it is correlated with various molecular quantities. The sum can be conveniently split into two parts, the external and the internal Wiener index, i.e., W=Wex+Win. The aim of this paper is to explore extremal properties of the former, Wex, which refers to the sum of distances between two vertices, of which at least one must be a leaf. More precisely, we prove that the maximum value of the external Wiener index amongst all graphs is attained by a balanced double broom, which gives an affirmative answer to a strengthened version of the conjecture proposed by Gutman, Furtula and Das in Gutman et al. (2016) by considering all graphs instead of considering only the class of trees. Finally, we discuss a potential extension of our work.
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