Distance Sequences to bound the Harary Index and other Wiener-type Indices of a Graph

TL;DR

This paper establishes new bounds on various Wiener-type indices of graphs, including the Harary and hyper-Wiener indices, using distance sequences, with implications for different classes of graphs.

Contribution

It introduces bounds on a broad class of distance-based indices, resolving open problems and providing sharp bounds for specific graph classes.

Findings

Sharp lower bounds on the Harary index for graphs of given order and size

Sharp upper bounds on the hyper-Wiener index for certain graph classes

Bounds applicable to trees with all vertices of odd degree

Abstract

In this paper we obtain bounds on a very general class of distance-based topological indices of graphs, which includes the Wiener index, defined as the sum of the distances between all pairs of vertices of the graph, and most generalisations of the Wiener index, including the Harary index and the hyper-Wiener index. Our results imply 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 for (i) graphs of given order and size (which resolves a problem in the monograph [The Harary index of a graph, Xu, Das, Trinajsti\'{c}, Springer (2015)], (ii) for $\kappa$-connected graphs, where $\kappa$ is even, (iii) for maximal outerplanar graphs and for Apollonian networks (a subclass of maximal planar graphs), and (iv) for trees in which all vertices have odd degree.