Five results on maximizing topological indices in graphs

Abstract

In this paper, we prove a collection of results on graphical indices. Wedetermine the extremal graphs attaining the maximal generalized Wiener index(e.g. the hyper-Wiener index) among all graphs with given matching number orindependence number. This generalizes some work of Dankelmann, as well as somework of Chung. We also show alternative proofs for two recents results onmaximizing the Wiener index and external Wiener index by deriving it fromearlier results. We end with proving two conjectures. We prove that the maximumfor the difference of the Wiener index and the eccentricity is attained by thepath if the order $n$ is at least $9$ and that the maximum weighted Szegedindex of graphs of given order is attained by the balanced complete bipartitegraphs.Comment: 13 pages, 4 figure