On Normalized Laplacian, Degree-Kirchhoff Index of the Strong Prism of Generalized Phenylenes^*

References

• J. A. Bondy, and U. S. R. Murty, Graph Theory with Applications (Mcmillan: London, 1976).
   Google Scholar
• H. Wiener, “Structural Determination of Paraffin Boiling Points,”Journal of the American Chemical Society 69, no. 1 (1947): 17–20.
   PubMed Web of Science ®Google Scholar
• I. Gutman, “Selected Properties of the Schultz Molecular Topological Index,”Journal of Chemical Information and Computer Sciences 34, no. 5 (1994): 1087–9.
   Google Scholar
• D. J. Klein, and M. Randić, “Resistance Distance,”Journal of Mathematical Chemistry 12, no. 1 (1993): 81–95.
   Web of Science ®Google Scholar
• I. Gutman, and B. Mohar, “The quasi-Wiener and the Kirchhoff Indices Coincide,”Journal of Chemical Information and Computer Sciences 36, no. 5 (1996): 982–5.
   Google Scholar
• F. R. K. Chung, Spectral Graph Theory (American Mathematical Society: Providence, RI, 1997).
   Google Scholar
• H. Y. Chen, and F. J. Zhang, “Resistance Distance and the Normalized Laplacian Spectrum,”Discrete Applied Mathematics 155, no. 5 (2007): 654–61.
   Web of Science ®Google Scholar
• L. H. Feng, I. Gutman, and G. H. Yu, “Degree Kirchhoff Index of Unicyclic Graphs,” MATCH Communications in Mathematical and in Computer Chemistry 69 (2013): 629–48.
   Web of Science ®Google Scholar
• J. Huang, and S. C. Li, “On the Normalised Laplacian Spectrum, degree-Kirchhoff Index and Spanning Trees of Graphs,” Bulletin of the Australian Mathematical Society 91, no. 3 (2015): 353–67.
   Web of Science ®Google Scholar
• C. He, S. C. Li, W. Luo, and L. Sun, “Calculating the Normalized Laplacian Spectrum and the Number of Spanning Trees of Linear Pentagonal Chains,” Journal of Computational and Applied Mathematics 344 (2018) : 381–93.
   Web of Science ®Google Scholar
• H. P. Zhang, and Y. J. Yang, “Resistance Distance and Kirchhoff Index in Circulant Graphs,”International Journal of Quantum Chemistry 107, no. 2 (2007): 330–9.
   Web of Science ®Google Scholar
• D. Q. Li and Y. P. Hou, “The Normalized Laplacian Spectrum of Quadrilateral Graphs and Its Applications,” Applied Mathematics and Computation 297 (2017) : 180–8.
   Web of Science ®Google Scholar
• D. J. Klein, I. Lukovits, and I. Gutman, “On the Definition of the hyper-Wiener Index for Cycle-Containing Structures,” Journal of Chemical Information and Computer Sciences 35, no. 1 (1995): 50–2.
   Google Scholar
• H. P. Zhang, Y. J. Yang and C. W. Li, “Kirchhoff Index of Composite Graphs,”Discrete Applied Mathematics 157,no. 13 (2009): 2918–27.
   Web of Science ®Google Scholar
• R. A. Bapat, M. Karimi, and J. B. Liu, “Kirchhoff Index and Degree Kirchhoff Index of Complete Multipartite Graphs,” Discrete Applied Mathematics 232 (2017): 41–9.
   Web of Science ®Google Scholar
• E. Bendito, A. Carmona, A. M. Encinas, and J. M. Gesto, “A Formula for the Kirchhoff Index,”International Journal of Quantum Chemistry 108, no. 6 (2008): 1200–6.
   Web of Science ®Google Scholar
• J. Huang, S. C. Li, and L. Q. Sun, “The Normalized Laplacians, degree-Kirchhoff Index and the Spanning Trees of Linear Hexagonal Chains,” Discrete Applied Mathematics 207 (2016): 67–79.
   Web of Science ®Google Scholar
• Y. X. Li, S. A. Xu, H. Hua, and X. F. Pan, “On the Resistance Diameter of the Cartesian and Lexicographic Product of Paths,” Journal of Applied Mathematics and Computing. Advance online publication. doi: 10.1007/s12190-021-01587-w.
   Web of Science ®Google Scholar
• S. C. Li, W. Wei, and S. Yu, “On Normalized Laplacians, Multiplicative Degree-Kirchhoff Indices and Spanning Trees of the Linear [n]Phenylenes and Their Dicyclobutadieno Derivatives,” International Journal of Quantum Chemistry 119, no. 8 (2019): e25863.
   Web of Science ®Google Scholar
• J. B. Liu, Z. Y. Shi, Y. H. Pan, J. Cao, M. A. Aty, and U. A. Juboori, “Computing the Laplacian Spectrum of Linear Octagonal-Quadrilateral Networks and Its Applications,” Polycyclic Aromatic Compounds.Advance online publication. doi: 10.1080/10406638.2020.1748666.
   Web of Science ®Google Scholar
• J. B. Liu, Q. Zheng, Z. Q. Cai, and S. Hayat, “On the Laplacians and Normalized Laplacians for Graph Transformation with Respect to the Dicyclobutadieno Derivative of [n]Phenylenes,” Polycyclic Aromatic Compounds. Advance online publication. doi: 10.1080/10406638.2020.1781209.
   Web of Science ®Google Scholar
• X. Ma, and H. Bian, “The Normalized Laplacians, degree-Kirchhoff Index and the Spanning Trees of Cylinder Phenylene Chain,” Polycyclic Aromatic Compounds 41, no. 6 (2021): 1159–79.
   Web of Science ®Google Scholar
• Y. G. Pan, and J. P. Li, “Kirchhoff Index, Multiplicative degree-Kirchhoff Index and Spanning Trees of the Linear Crossed Hexagonal Chains,”International Journal of Quantum Chemistry 118,no. 24 (2018): e25787.
   Web of Science ®Google Scholar
• J. Palacios, and J. M. Renom, “Another Look at the degree-Kirchhoff Index,” International Journal of Quantum Chemistry 111, (2011):3453–5.
   Web of Science ®Google Scholar
• Y. Peng, and S. Li, “On the Kirchhoff Index and the Number of Spanning Trees of Linear Phenylenes,” MATCH Communications in Mathematical and in Computer Chemistry 77, no. 3 (2017): 765–80.
   Web of Science ®Google Scholar
• L. Pavlovic’ and I. Gutman, “Wiener Numbers of Phenylenes: An Exact Result,” Journal of Chemical Information and Computer Sciences 37, (1997): 355–8.
   Google Scholar
• A. L. Chen, and F. J. Zhang, “Wiener Index and Perfect Matchings in Random Phenylene Chains,” MATCH Communications in Mathematical and in Computer Chemistry 61 (2009): 623–30.
   Web of Science ®Google Scholar
• Z. X. Zhu, and J. B. Liu, “The Normalized Laplacian, degree-Kirchhoff Index and the Spanning Tree Numbers of Generalized Phenylenes,” Discrete Applied Mathematics 254, (2019) : 256–67.
   Web of Science ®Google Scholar
• Z. Li, Z. Xie, J. Li, and Y. Pan, “Resistance Distance-Based Graph Invariants and Spanning Trees of Graphs Derived from the Strong Prism of a Star,” Applied Mathematics and Computation 382 (2020) : 125335.
   Web of Science ®Google Scholar
• Y. G. Pan, and J. P. Li, “Resistance distance-based graph invariants and spanning trees of graphs derived from the strong product of P2 and Cn,” (2019), arXiv:1906.04339.
   Google Scholar
• C. Liu, Y. Pan, and J. Li, “On the Laplacian Spectrum and Kirchhoff Index of Generalized Phenylenes,” Polycyclic Aromatic Compounds. Advance online publication. doi: 10.1080/10406638.2019.1703765.
   Web of Science ®Google Scholar
• Y. Pan, C. Liu, and J. Li, “Kirchhoff Indices and Numbers of Spanning Trees of Molecular Graphs Derived from Linear Crossed Polyomino Chain,” Polycyclic Aromatic Compounds.Advance online publication. doi: 10.1080/10406638.2020.1725898.
   Web of Science ®Google Scholar
• J. Huang, S. C. Li, and X. C. Li, “The Normalized Laplacian degree-Kirchhoff Index and Spanning Trees of the Linear Polyomino Chains,” Applied Mathematics and Computation 289 (2016): 324–34.
   Web of Science ®Google Scholar