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

Abstract As a powerful tool for describing and studying the properties of compounds, the graphs spectrum analysis and calculations have attracted substantial attention of the scientific community. Let On denote linear octagonal-quadrilateral networks. In this paper, we investigate that the Laplacian spectrum of On consists of the Laplacian spectrum of and eigenvalues of a symmetric tridiagonal matrix of order As applications of the obtained results, we derive the explicit closed formulas of Kirchhoff index and complexity of On on the basis of the relationship between the coefficients and roots of the characteristic polynomial.

[1]  Jing Huang,et al.  The normalized Laplacians, degree-Kirchhoff index and the spanning trees of linear hexagonal chains , 2016, Discret. Appl. Math..

[2]  Jia-Bao Liu,et al.  The Complexity of Some Classes of Pyramid Graphs Created from a Gear Graph , 2018, Symmetry.

[3]  Xianya Geng,et al.  On the Kirchhoff Index and the Number of Spanning Trees of Linear Phenylenes Chain , 2021, Polycyclic Aromatic Compounds.

[4]  Keivan Hassani Monfared,et al.  An analog of Matrix Tree Theorem for signless Laplacians , 2018, Linear Algebra and its Applications.

[5]  Shuchao Li,et al.  Calculating the normalized Laplacian spectrum and the number of spanning trees of linear pentagonal chains , 2018, J. Comput. Appl. Math..

[6]  Y. Pan,et al.  On the Laplacian Spectrum and Kirchhoff Index of Generalized Phenylenes , 2019, Polycyclic Aromatic Compounds.

[7]  Y. Pan,et al.  Kirchhoff index, multiplicative degree-Kirchhoff index and spanning trees of the linear crossed hexagonal chains , 2018, International Journal of Quantum Chemistry.

[8]  Zhongxun Zhu,et al.  On the number of spanning trees and normalized Laplacian of linear octagonal‐quadrilateral networks , 2019, International Journal of Quantum Chemistry.

[9]  Jing Huang,et al.  The normalized Laplacians on both k-triangle graph and k-quadrilateral graph with their applications , 2018, Appl. Math. Comput..

[10]  Y. Pan,et al.  Kirchhoff Indices and Numbers of Spanning Trees of Molecular Graphs Derived from Linear Crossed Polyomino Chain , 2020, Polycyclic Aromatic Compounds.

[11]  Yuliang Yang,et al.  Graph theory of viscoelasticities for polymers with starshaped, multiple‐ring and cyclic multiple‐ring molecules , 1985 .

[12]  Heping Zhang,et al.  Kirchhoff index of linear hexagonal chains , 2008 .

[13]  Yongqiang Cheng,et al.  Patient-Specific Coronary Artery 3D Printing Based on Intravascular Optical Coherence Tomography and Coronary Angiography , 2019, Complex..

[14]  Hong Bian,et al.  The normalized Laplacians, degree-Kirchhoff index and the spanning trees of hexagonal Möbius graphs , 2019, Appl. Math. Comput..

[15]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[16]  Shuchao Li,et al.  On the normalized Laplacian of Möbius phenylene chain and its applications , 2019, International Journal of Quantum Chemistry.

[17]  D. Klein,et al.  Graph Cyclicity, Excess Conductance, and Resistance Deficit , 2001 .

[18]  Jia-Bao Liu,et al.  Number of Spanning Trees in the Sequence of Some Graphs , 2019, Complex..

[19]  Roger C. Entringer,et al.  Distance in graphs , 1976 .

[20]  Shujing Wang,et al.  On the Kirchhoff Indices and the Number of Spanning Trees of Möbius Phenylenes Chain and Cylinder Phenylenes Chain , 2019, Polycyclic Aromatic Compounds.

[21]  Zhongxun Zhu,et al.  The normalized Laplacian, degree-Kirchhoff index and the spanning tree numbers of generalized phenylenes , 2019, Discret. Appl. Math..

[22]  Bojan Mohar,et al.  The Quasi-Wiener and the Kirchhoff Indices Coincide , 1996, J. Chem. Inf. Comput. Sci..

[23]  T. D. Morley,et al.  Eigenvalues of the Laplacian of a graph , 1985 .

[24]  H. Bian,et al.  The Normalized Laplacians, Degree-Kirchhoff Index and the Spanning Trees of Cylinder Phenylene Chain , 2019, Polycyclic Aromatic Compounds.

[25]  Jing Huang,et al.  The normalized Laplacian, degree-Kirchhoff index and spanning trees of the linear polyomino chains , 2016, Appl. Math. Comput..

[26]  Shuchao Li,et al.  Multiplicative degree‐Kirchhoff index and number of spanning trees of a zigzag polyhex nanotube TUHC [2 n , 2] , 2019, International Journal of Quantum Chemistry.

[27]  Douglas J. Klein,et al.  Resistance-Distance Sum Rules* , 2002 .

[28]  Shuchao Li,et al.  On normalized Laplacians, multiplicative degree‐Kirchhoff indices, and spanning trees of the linear [n]phenylenes and their dicyclobutadieno derivatives , 2018, International Journal of Quantum Chemistry.

[29]  J. Zhao,et al.  Resistance distance-based graph invariants and the number of spanning trees of linear crossed octagonal graphs , 2019, J. Appl. Math. Comput..

[30]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[31]  Jing Zhao,et al.  On the generalized adjacency, Laplacian and signless Laplacian spectra of the weighted edge corona networks , 2020 .

[32]  István Lukovits,et al.  Extensions of the Wiener Number , 1996, J. Chem. Inf. Comput. Sci..