Linear Algorithms for the Hosoya Index and Hosoya Matrix of a Tree

Aleksander Vesel

doi:10.3390/math9020142

Abstract

The Hosoya index of a graph is defined as the total number of its independent edge sets. This index is an important example of topological indices, a molecular-graph based structure descriptor that is of significant interest in combinatorial chemistry. The Hosoya index inspires the introduction of a matrix associated with a molecular acyclic graph called the Hosoya matrix. We propose a simple linear-time algorithm, which does not require pre-processing, to compute the Hosoya index of an arbitrary tree. A similar approach allows us to show that the Hosoya matrix can be computed in constant time per entry of the matrix.

Highlights

In Haruo Hosoya’s seminal paper, a molecular-graph based structure descriptor is proposed that nowadays is known under the name Hosoya index [1]
It is well-known that several physicochemical properties of chemical structures are well correlated with the Hosoya index of the corresponding molecular graphs
We have obtained efficient algorithms for computing the Hosoya index and the Hosoya matrix on an arbitrary acyclic graph

Summary

Introduction

In Haruo Hosoya’s seminal paper, a molecular-graph based structure descriptor is proposed that nowadays is known under the name Hosoya index [1]. It is well-known that several physicochemical properties of chemical structures are well correlated with the Hosoya index of the corresponding molecular graphs. Using this result, we present a linear-time algorithm for computing the Hosoya index of a graph of this class. The results allow the computation of the Hosoya matrix in constant time per entry of the matrix

Preliminaries

Algorithm for Computing the Hosoya Index

Example

Algorithm for Computing the Hosoya Matrix

Conclusions