Additively weighted Harary index of some composite graphs
type: Item
instance of: scholarly article
title: Additively weighted Harary index of some composite graphs
zbMATH Open document ID: 1254.05191
author: Yaser Alizadeh
author: A. Iranmanesh
author: Tomislav Došlić
published in: Discrete Mathematics
publication date: 2012-11-22T00:00:00Z
review text: The paper under review introduces a modification of the Harary index of a connected graph. The Harary index of a connected graph \(G=(V,E)\) is \[ H(G)=\sum_{\{u,v\}\in V^{[2]}}\frac{1}{d(u,v)} \] where \(d(u,v)\) denotes the graph distance between the two vertices \(u\neq v\in V\). Such indices are sometimes applied in chemistry. For example, the Harary index of a single edge is 1. The aim of the new invariant is to give extra weight to vertices with large degree. Specifically, the paper studies the additively weighted Harary index \[ H_{A}(G)=\sum _{\{u,v\}\in V^{[2]}}\frac{d(u)+d(v)}{d(u,v)}. \] Here \(d(w)\) is the degree of the vertex \(w\). Thus, letting \(K_{n}\) and \(S_{n}\) denote respectively a complete graph and a star on \(n\) vertices, \(H_{A}(K_{n})=n(n-1)^{2}\) and \(H_{A}(S_{n})=(n-1)(5n-2)/4\). It is clear that adding an edge can only increase \(H_{A}\), so the value for the complete graph is the maximum attainable: similarly a tree must have the smallest value. The authors use transformations on trees to show that amongst trees on \(n\) vertices, the path has the smallest \(H_{A}\) and the star the largest. They then discuss various ``composite graphs'', i.e. graphs obtained by putting two other graphs together in some way. They obtain various formulae for the additively weighted Harary indices of various graphs in terms of additively weighted Harary indices of the two input graphs and the so-called Zagreb indices of the graphs. Some future directions for research are also indicated.
reviewed by: David B. Penman
Mathematics Subject Classification ID: 05C90
Mathematics Subject Classification ID: 05C12
Mathematics Subject Classification ID: 05C07
Mathematics Subject Classification ID: 92E10
zbMATH DE Number: 6109888
zbMATH Keywords: additively weighted Harary index
zbMATH Keywords: join
zbMATH Keywords: composition
zbMATH Keywords: disjunction
zbMATH Keywords: symmetric difference
MaRDI profile type: Publication
full work available at URL: j.disc.2012.09.011
OpenAlex ID: W2087600287
DOI: 10.1016/J.DISC.2012.09.011
description: scientific article; zbMATH DE number 6109888