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

Tu, Yifeng; Ma, Xuanlong; Zhang, Yuqing; Ren, Junyu

doi:10.3390/axioms11060278

Cited by 2 publications

(4 citation statements)

References 23 publications

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“…Based on Claims 1, 3 and Lemma 3, we can get the same results as the Theorem 3 [29], which further proves that the result of our calculation (Theorem 2) is correct. Theorem 2.…”

Section: Degree-kirchhoff Index and The Number Of Spanning Trees Of Qp Nsupporting

confidence: 71%

“…Then the linear pentagonal derivation chain, denoted by QP n , is the graph obtained by attaching four-membered rings to each hexagon composed of two pentagons of P n , as showed in Figure 1. It is not difficult to verify that |V(QP The explicit closed-form formulas for Kirchhoff index and the number of spanning trees of the linear pentagonal derivation chain QP n have been derived from the Laplacian spectrum [29]. Motivated by the above works, we consider the degree-Kirchhoff index and the number of spanning trees of linear pentagonal derivation chain in terms of the normalized Laplacian spectrum.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by the above works, we consider the degree-Kirchhoff index and the number of spanning trees of linear pentagonal derivation chain in terms of the normalized Laplacian spectrum. Different from the method in [29], in this paper, we solve the number of spanning trees according to the normalized Laplacian spectrum, which gives a new way for calculating the number of spanning trees of QP n .…”

Section: Introductionmentioning

confidence: 99%

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On the Normalized Laplacian Spectrum of the Linear Pentagonal Derivation Chain and Its Application

Zhang,

2023

Axioms

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A novel distance function named resistance distance was introduced on the basis of electrical network theory. The resistance distance between any two vertices u and v in graph G is defined to be the effective resistance between them when unit resistors are placed on every edge of G. The degree-Kirchhoff index of G is the sum of the product of resistance distances and degrees between all pairs of vertices of G. In this article, according to the decomposition theorem for the normalized Laplacian polynomial of the linear pentagonal derivation chain QPn, the normalize Laplacian spectrum of QPn is determined. Combining with the relationship between the roots and the coefficients of the characteristic polynomials, the explicit closed-form formulas for degree-Kirchhoff index and the number of spanning trees of QPn can be obtained, respectively. Moreover, we also obtain the Gutman index of QPn and we discovery that the degree-Kirchhoff index of QPn is almost half of its Gutman index.

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“…Based on Claims 1, 3 and Lemma 3, we can get the same results as the Theorem 3 [29], which further proves that the result of our calculation (Theorem 2) is correct. Theorem 2.…”

Section: Degree-kirchhoff Index and The Number Of Spanning Trees Of Qp Nsupporting

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Normalized Laplacian Spectrum of the Linear Pentagonal Derivation Chain and Its Application

Zhang,

2023

Axioms

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“…In their work [14], Feng et al investigated the Kirchhoff index of phenylenes. 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.…”

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

Cited by 2 publications

References 23 publications

On the Normalized Laplacian Spectrum of the Linear Pentagonal Derivation Chain and Its Application

On the Normalized Laplacian Spectrum of the Linear Pentagonal Derivation Chain and Its Application

Kirchhoff Index and Degree Kirchhoff Index of Tetrahedrane-Derived Compounds

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