Wiener–Hosoya Matrix of Connected Graphs

Hassan Ibrahim,Reza Sharafdini,Tamás Réti,Abolape Akwu

doi:10.3390/math9040359

Hassan Ibrahim, Reza Sharafdini + Show 2 more

Open Access

https://doi.org/10.3390/math9040359

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Abstract

Let G be a connected (molecular) graph with the vertex set V(G)={v1,⋯,vn}, and let di and σi denote, respectively, the vertex degree and the transmission of vi, for 1≤i≤n. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the n×n matrix whose (i,j)-entry is equal to σi2di+σj2dj if vi and vj are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.

Highlights

Several different topological indices and other molecular descriptors derived from them like energy and spectral radius, have been studied so far and have been used in Quantitative Structure Activity Relationship (QSAR)/Quantitative Structure PropertyRelationship (QSPR) studies
We are concerned with the Wiener index which is a most celebrated distance-based topological index, defined as the sum of distances between vertices of a connected graph
The Wiener index of a connected graph was the first topological index to be used in chemistry

Summary

Introduction

Several different topological indices and other molecular descriptors derived from them like energy and spectral radius, have been studied so far and have been used in Quantitative Structure Activity Relationship (QSAR)/Quantitative Structure Property. An standard way of computing and describing some specific topological indices is to associate an extended adjacency matrix to them and investigate the other molecular descriptors derived from them, for example eigenvalues and energy This technique has been applied to both degree-based (See [6,7,8] and references therein) and distance-based. The transmission of a vertex vi ∈ V ( G ), denoted by σi = σG (vi ), is defined as the sum of distances between vi and any other vertices in G, i.e., n σi = σG (vi ) =. A non-negative matrix A = ( ai,j ) is said to be irreducible, if the directed graph ΓA with the vertex set {1, . Let A be an n × n, non-negative, symmetric matrix with eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn. The equality holds if and only if x is an eigenvector of A corresponding to λ1

Relation with Adjacency Matrix

Bounds on Wiener–Hosoya Spectral Radius of Connected Graphs

Concluding Remarks