Peripheral hyper-Wiener index of a graph

TL;DR

This paper introduces the peripheral hyper-Wiener index, a new topological graph index extending the peripheral Wiener index, and explores its properties, formulas for specific graph classes, and bounds relating to graph parameters.

Contribution

It defines the peripheral hyper-Wiener index, derives explicit formulas for certain graphs, and establishes bounds, extending previous work on the peripheral Wiener index.

Findings

Explicit formula for hypercubes

Bounds on PWW(G) in terms of graph parameters

Analogues of previous results for PW(G)

Abstract

In this note, we introduce a new topological index of a graph G that we term peripheral hyper-Wiener index, denoted PWW(G). It is a natural extension of the peripheral Wiener index PW(G) initiated in [NB17] and is to the peripheral Wiener index what the hyper-Wiener index is to the Wiener index. We investigate its basic properties. We compute the peripheral hyper-Wiener index of the cartesian product and trees. In particular, we get an explicit formula for the case of the hypercubes. We also give lower and upper bounds on PW(G) and PWW(G) in terms of the order, size, diameter and the number of peripheral vertices. This paper is an echo to [NB17], most of the results we get are analogues of the ones therein.