Comparison of the Wiener and Kirchhoff Indices of Random Pentachains

Wei, Shouliu; Shiu, Wai Chee; Ke, Xiaoling; Jian-wu, Huang

doi:10.1155/2021/7523214

Cited by 2 publications

(1 citation statement)

References 44 publications

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“…In another work [15], they studied the Kirchhoff index and the number of spanning trees of linear pentagonal derivation chains. Meanwhile the authors of [16] compared the winner index and Kirchhoff index of random pentane chains and obtained some conclusions. Yang studied the upper and lower bounds of the Kirchhoff index of planar graphs and fullerene graphs in the literature [17].…”

Section: Introductionmentioning

confidence: 99%

Kirchhoff Index and Degree Kirchhoff Index of Tetrahedrane-Derived Compounds

Zhao

Wang

et al. 2023

Symmetry

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Tetrahedrane-derived compounds consist of n crossed quadrilaterals and possess complex three-dimensional structures with high symmetry and dense spatial arrangements. As a result, these compounds hold great potential for applications in materials science, catalytic chemistry, and other related fields. The Kirchhoff index of a graph G is defined as the sum of resistive distances between any two vertices in G. This article focuses on studying a type of tetrafunctional compound with a linear crossed square chain shape. The Kirchhoff index and degree Kirchhoff index of this compound are calculated, and a detailed analysis and discussion is conducted. The calculation formula for the Kirchhoff index is obtained based on the relationship between the Kirchhoff index and Laplace eigenvalue, and the number of spanning trees is derived for linear crossed quadrangular chains. The obtained formula is validated using Ohm’s law and Cayley’s theorem. Asymptotically, the ratio of Kirchhoff index to Wiener index approaches one-fourth. Additionally, the expression for the degree Kirchhoff index of the linear crossed quadrangular chain is obtained through the relationship between the degree Kirchhoff index and the regular Laplace eigenvalue and matrix decomposition theorem.

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Section: Introductionmentioning

confidence: 99%

Kirchhoff Index and Degree Kirchhoff Index of Tetrahedrane-Derived Compounds

Zhao

Wang

et al. 2023

Symmetry

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On the Laplacian, the Kirchhoff Index, and the Number of Spanning Trees of the Linear Pentagonal Derivation Chain

Zhang

et al. 2022

Axioms

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Let Pn be a pentagonal chain with 2n pentagons in which two pentagons with two edges in common can be regarded as adding one vertex and two edges to a hexagon. Thus, the linear pentagonal derivation chains QPn represent the graph obtained by attaching four-membered rings to every two pentagons of Pn. In this article, the Laplacian spectrum of QPn consisting of the eigenvalues of two symmetric matrices is determined. Next, the formulas for two graph invariants that can be represented by the Laplacian spectrum, namely, the Kirchhoff index and the number of spanning trees, are studied. Surprisingly, the Kirchhoff index is almost one half of the Wiener index of a linear pentagonal derivation chain QPn.

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Comparison of the Wiener and Kirchhoff Indices of Random Pentachains

Abstract: Let G be a connected (molecule) graph. The Wiener index W G and Kirchhoff index K f … Show more

Cited by 2 publications

References 44 publications

Kirchhoff Index and Degree Kirchhoff Index of Tetrahedrane-Derived Compounds

Kirchhoff Index and Degree Kirchhoff Index of Tetrahedrane-Derived Compounds

On the Laplacian, the Kirchhoff Index, and the Number of Spanning Trees of the Linear Pentagonal Derivation Chain

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