On the Definition of the Hyper-Wiener Index for Cycle-Containing Structures

Abstract

The hyper-Wiener index was recently introduced by Randic. The original definition given by Randié can be used for acyclic structures only. In this paper the definition of Randic was extended in two different fashions so as to be applicable for any connected structure. The formula provides an easy method to calculate the hyper-Wiener index for any graph.

Entropy Prediction of Benzene Derivatives Using Topological Indices

“…where d(u,v) denotes the distance between the vertices u and v in the graph G and the summations run over all pairs of vertices of G. Randić's original definition (1993) [27] of the hyper-Wiener index is applicable to trees only.…”

Introdução à Teoria Espectral de Grafos com Aplicações

“…Na literatura de Química, o maior autovalor da matriz distância funciona como um descritor da estrutura deárvores e parece inferir sobre a extensão de ramificação e de pontos de ebulição de alcanos, quando estes são modelados porárvores, [3] e [78]. Além disso, o importanteíndice de Wiener, que veremos aqui,é definido a partir da matriz distância de um grafo e há inúmeros artigos publicados nesta direção, [3], [4], [35], [54], [55], [62], [66], [68] e [78]. …”

On Graphs with Same Distance Distribution

“…As an extension of the Wiener index of a tree, Randić [2] [5] generalized the hyper-Wiener index so as to be applicable to any connected structure. Their formula for the hyper-Wiener index R of a graph G is…”