In this paper we obtain several new bounds on well-studied topological indices, among those sharp lower bounds on the Harary index and sharp upper bounds on the hyper-Wiener index and the multiplicative Wiener index for (i) graphs of given order and size (which solves a problem in the monograph [Xu, Das, Trinajstić, The Harary index of a graph, Springer (2015)]), (ii) $\kappa$-connected graphs, where \( \kappa \) is even, (iii) maximal outerplanar graphs, (iv) Apollonian networks, and (v) for trees in which all vertices have odd degree.

We use a novel approach to prove our bounds for a very general class of distance-based topological indices of graphs, which includes the Wiener index and most of its generalizations, including the Harary index, the hyper-Wiener index and the multiplicative Wiener index.