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Hosoya Polynomial for Subdivided Caterpillar Graphs.

Muhammad Numan1, Aamra Nawaz1, Adnan Aslam2

  • 1Department of Mathematics, COMSATS University Islamabad Attock Campus , Attock,Pakistan.

Combinatorial Chemistry & High Throughput Screening
|December 14, 2020
PubMed
Summary
This summary is machine-generated.

This study provides a closed-form formula for the Hosoya polynomial of subdivided caterpillar trees. This enables easier computation of Wiener and Hyper Wiener indices for molecular structures.

Keywords:
Hosoya polynomialcaterpillar graphdiameter.hyper-wiener indexuniform subdivisionwiener index

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Area of Science:

  • Chemical graph theory
  • Mathematical chemistry

Background:

  • The Hosoya polynomial is crucial for calculating topological indices like the Wiener index.
  • These indices reveal molecular topology features (branching, cyclicity, centricity) influencing physicochemical properties.
  • Caterpillar trees model benzenoid hydrocarbon structures in chemical graph theory.

Purpose of the Study:

  • To compute the closed-form formula for the Hosoya polynomial of subdivided caterpillar trees.
  • To facilitate the calculation of Wiener and Hyper Wiener indices for these graph structures.

Main Methods:

  • The Hosoya polynomial is defined as H(G;x) = Σd(G.k)x^k, where d(G.k) is the number of vertex pairs at distance k.
  • Coefficients d(G.m) are determined by classifying and counting vertex pairs at specific distances.
  • This method yields the Hosoya polynomial for uniform subdivision of caterpillar graphs.

Main Results:

  • A closed-form formula for the Hosoya polynomial of subdivided caterpillar trees has been successfully computed.
  • This formula simplifies the calculation of the Wiener index and Hyper Wiener index for uniform subdivision of caterpillar graphs.

Conclusions:

  • Subdivided caterpillar trees are important graph classes with applications in chemical graph theory.
  • This research offers a methodological framework for studying Hosoya polynomials in various tree structures.
  • The findings contribute to a better understanding of the relationship between molecular structure and properties.