On the Kirchhoff Indices and the Number of Spanning Trees of Möbius Phenylenes Chain and Cylinder Phenylenes Chain

Geng, Xianya; Wang, Ping; Lei, Lan; Wang, Shujing

doi:10.1080/10406638.2019.1693405

Cited by 22 publications

(18 citation statements)

References 20 publications

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“…Applying similar methods, X. Ma et al [26] determined the degree-Kirchhoff index and spanning trees of linear hexagonal Möbius graphs. For the results of linear phenylenes, see Reference [5,23]. Considering a more complex graph and different methods, the degree-Kirchhoff index and number of spanning trees of liner heptagonal networks were given in this paper.…”

Section: Discussionmentioning

confidence: 99%

“…Combining the adjacency and degree matrix, we get the Laplacian matrix, for which expressions can be written as L(G) = D(G) − A(G), respectively. One may deeply understand them with the help of Reference [3][4][5][6]. The normalized Laplacian [7] is given by:…”

Section: Introductionmentioning

confidence: 99%

“…constructed a crossed hexagonal by adding two pairs of crossed edges in linear hexagonal chains, and they got the Kirchhoff and degree-Kirchhoff index [21], respectively (see Reference [5,22,23] for details.) Let H n be the heptagonal networks as illustrated in Figure 1, of which two heptagons have two common edges.…”

Section: Introductionmentioning

confidence: 99%

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On the Normalized Laplacian and the Number of Spanning Trees of Linear Heptagonal Networks

et al. 2019

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The normalized Laplacian plays an important role on studying the structure properties of non-regular networks. In fact, it focuses on the interplay between the structure properties and the eigenvalues of networks. Let H n be the linear heptagonal networks. It is interesting to deduce the degree-Kirchhoff index and the number of spanning trees of H n due to its complicated structures. In this article, we aimed to first determine the normalized Laplacian spectrum of H n by decomposition theorem and elementary operations which were not stated in previous results. We then derived the explicit formulas for degree-Kirchhoff index and the number of spanning trees with respect to H n .

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Normalized Laplacian and the Number of Spanning Trees of Linear Heptagonal Networks

et al. 2019

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“…J. Huang et al [7] got the normalized Laplacian spectrum of linear hexagonal networks by using the decomposition theorem. Then, the Laplacian spectrum of linear phenylenes were derived [12]. Besides, Z. Zhu and J. Liu [17] obtained the Laplacian spectrum of generalized phenylenes.…”

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confidence: 99%

Computing the Laplacian Spectrum of Linear Octagonal-Quadrilateral Networks and Its Applications

Liu

Shi

Pan

et al. 2020

Polycyclic Aromatic Compounds

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“…It is more meaningful to determine the Kirchhoff or multiplicative degree-Kirchhoff index. Up to date, the Kirchhoff or multiplicative degree-Kirchhoff index have already determined of some classes of graphs, see [15][16][17][18][19][20][21]. In this paper, we only discuss a class of linear graphs.…”

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confidence: 99%

Resistance distance-based graph invariants and the number of spanning trees of linear crossed octagonal graphs

Zhao

Liu

Hayat

2019

J. Appl. Math. Comput.

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Resistance distance is a novel distance function, also a new intrinsic graph metric, which makes some extensions of ordinary distance. Let O n be a linear crossed octagonal graph. Recently, Pan and Li (2018) derived the closed formulas for the Kirchhoff index, multiplicative degree-Kirchhoff index and the number of spanning trees of H n . They pointed that it is interesting to give the explicit formulas for the Kirchhoff and multiplicative degree-Kirchhoff indices of O n . Inspired by these, in this paper, two resistance distance-based graph invariants, namely, Kirchhoff and multiplicative degree-Kirchhoff indices are studied. We firstly determine formulas for the Laplacian (normalized Laplacian, resp.) spectrum of O n . Further, the formulas for those two resistance distance-based graph invariants and spanning trees are given. More surprising, we find that the Kirchhoff (multiplicative degree-Kirchhoff, resp.) index is almost one quarter to Wiener (Gutman, resp.) index of a linear crossed octagonal graph.

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On the Kirchhoff Indices and the Number of Spanning Trees of Möbius Phenylenes Chain and Cylinder Phenylenes Chain

Cited by 22 publications

References 20 publications

On the Normalized Laplacian and the Number of Spanning Trees of Linear Heptagonal Networks

On the Normalized Laplacian and the Number of Spanning Trees of Linear Heptagonal Networks

Computing the Laplacian Spectrum of Linear Octagonal-Quadrilateral Networks and Its Applications

Resistance distance-based graph invariants and the number of spanning trees of linear crossed octagonal graphs

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