Analyzing the normalized Laplacian spectrum and spanning tree of the cross of the derivative of linear networks

Ze-Miao Dai; Jia-Bao Liu; Kang Wang; Ze-Miao Dai; Jia-Bao Liu; Kang Wang

doi:10.3934/math.2024710

AIMS Mathematics

2024, Volume 9, Issue 6: 14594-14617. doi: 10.3934/math.2024710

Previous Article Next Article

Research article

Analyzing the normalized Laplacian spectrum and spanning tree of the cross of the derivative of linear networks

1.
College of Information Technology, Anhui Vocational College of Defense Technology, Luan 237011, China
2.
School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China

Received: 20 December 2023 Revised: 19 March 2024 Accepted: 16 April 2024 Published: 23 April 2024
MSC : 05C50, 05C90

In this paper, we focus on the strong product of the pentagonal networks. Let $ R_{n} $ be a hexagonal network composed of $ 2n $ pentagons and $ n $ quadrilaterals. Let $ P_{n}^{2} $ denote the graph formed by the strong product of $ R_{n} $ and its copy $ R_{n}^{\prime} $. By utilizing the decomposition theorem of the normalized Laplacian characteristics polynomial, we characterize the explicit formula of the multiplicative degree-Kirchhoff index completely. Moreover, the complexity of $ P_{n}^{2} $ is determined.
- (multiplicative degree) Kirchhoff index,
- Wiener index,
- Gutman index,
- spanning trees
Citation: Ze-Miao Dai, Jia-Bao Liu, Kang Wang. Analyzing the normalized Laplacian spectrum and spanning tree of the cross of the derivative of linear networks[J]. AIMS Mathematics, 2024, 9(6): 14594-14617. doi: 10.3934/math.2024710

Related Papers:

Abstract

In this paper, we focus on the strong product of the pentagonal networks. Let $ R_{n} $ be a hexagonal network composed of $ 2n $ pentagons and $ n $ quadrilaterals. Let $ P_{n}^{2} $ denote the graph formed by the strong product of $ R_{n} $ and its copy $ R_{n}^{\prime} $. By utilizing the decomposition theorem of the normalized Laplacian characteristics polynomial, we characterize the explicit formula of the multiplicative degree-Kirchhoff index completely. Moreover, the complexity of $ P_{n}^{2} $ is determined.

References

[1]	J. A. Bondy, U. S. R. Murty, Graph theory with applications, Macmillan Press Ltd., 1976.
[2]	F. R. K. Chung, Spectral graph theory, American Mathematical Society, 1997.
[3]	H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc., 69 (1947), 17–20. https://doi.org/10.1021/ja01193a005 doi: 10.1021/ja01193a005
[4]	A. Dobrynin, Branchings in trees and the calculation of the Wiener index of a tree, MATCH Commun. Math. Comput. Chem., 41 (2000), 119–134.
[5]	R. C. Entringer, D. E. Jackson, D. A. Snyder, Distance in graphs, Czechoslovak Mathematical Journal, 26 (1976), 283–296.
[6]	L. H. Feng, X. M. Zhu, W. J. Liu, Wiener index, Harary index and graph properties, Discrete Appl. Math., 223 (2017), 72–83. https://doi.org/72-83.10.1016/j.dam.2017.01.028
[7]	A. Abiad, B. Brimkov, A. Erey, On the Wiener index, distance cospectrality and transmission-regular graphs, Discrete Appl. Math., 230 (2017), 1–10. https://doi.org/10.1016/j.dam.2017.07.010 doi: 10.1016/j.dam.2017.07.010
[8]	Y. P. Mao, Z. Wang, I. Gutman, Nordhaus-Gaddum-type results for the Steiner Wiener index of graphs, Discrete Appl. Math., 219 (2017), 167–175. https://doi.org/167-175.10.1016/j.dam.2016.11.014
[9]	A. R. Ashrafi, A. Ghalavand, Ordering chemical trees by Wiener polarity index, Appl. Math. Comput., 313 (2017), 301–312. https://doi.org/301-312.10.1016/j.amc.2017.06.005
[10]	M. Crepnjak, N. Tratnik, The Szeged index and the Wiener index of partial cubes with applications to chemical graphs, Appl. Math. Comput., 309 (2017), 324–333. https://doi.org/324-333.10.1016/j.amc.2017.04.011
[11]	A. Mohajeri, P. Manshour, M. Mousaee, A novel topological descriptor based on the expanded Wiener index: applications to QSPR/QSAR studies, Iran. J. Math. Chem., 8 (2017), 107–135.
[12]	I. Gutman, Selected properties of the schultz molecular topological index, Journal of Chemical Information and Computer Sciences, 34 (1994), 1087–1089.
[13]	D. J. Klein, Resistance-distance sum rules, Croat. Chem. Acta, 75 (2002), 633–649.
[14]	D. J. Klein, O. Ivanciuc, Graph cyclicity, excess conductance, and resistance deficit, J. Math. Chem., 30 (2001), 271–287. https://doi.org/10.1023/A:1015119609980
[15]	H. Y. Chen, F. J. Zhang, Resistance distance and the normalized Laplacian spectrum, Discrete Appl. Math., 155 (2007), 654–661. https://doi.org/10.1016/j.dam.2006.09.008 doi: 10.1016/j.dam.2006.09.008
[16]	L. H. Feng, I. Gutman, G. H. Yu, Degree Kirchhoff index of unicyclic graphs, Match (2013).
[17]	J. Huang, S. H. Li, On the normalised Laplacian spectrum, degree-Kirchhoff index and spanning trees of graphs, B. Aust. Math. Soc., 91 (2015), 353–367. https://doi.org/10.1017/S0004972715000027 doi: 10.1017/S0004972715000027
[18]	Y. L. Yang, T. Y. Yu, Graph theory of viscoelasticities for polymers with starshaped, multiple-ring and cyclic multiple-ring molecules, Macromol. Chem. Phys., 186 (1985), 609–631.
[19]	G. H. Yua, L. H. Feng, On connective eccentricity index of graphs, MATCH Commun. Math. Comput. Chem., 69 (2013), 611–628.
[20]	X. L. Ma, B. Hong, The normalized Laplacians, degree-Kirchhoff index and the spanning trees of cylinder phenylene chain, Polycyclic Aromatic Compounds, 41 (2021), 1159–1179. https://doi.org/10.1080/10406638.2019.1665553 doi: 10.1080/10406638.2019.1665553
[21]	L. Lei, X. Y. Geng, S. Li, Y. Peng, Y. Yu, On the normalized Laplacian of Mobius phenylene chain and its applications, Int. J. Quantum Chem., 119 (2019), e26044. https://doi.org/10.1002/qua.26044 doi: 10.1002/qua.26044
[22]	U. Ali, Y. Ahmad, S. A. Xu, X. F. Pan, On Normalized Laplacian, Degree-Kirchhoff Index of the Strong Prism of Generalized Phenylenes, Polycyclic Aromatic Compounds, 42 (2022), 6215–6232. https://doi.org/10.1080/10406638.2021.1977351 doi: 10.1080/10406638.2021.1977351
[23]	X. Y. Geng, L. Yu, On the Kirchhoff index and the number of spanning trees of linear phenylenes chain, Polycyclic Aromatic Compounds, 42 (2022), 4984–4993. https://doi.org/10.1080/10406638.2021.1923536 doi: 10.1080/10406638.2021.1923536
[24]	J. B. Liu, Z. Y. Shi, Y. H. Pan, J. D. Cao, Computing the Laplacian spectrum of linear octagonal-quadrilateral networks and its applications, Polycyclic Aromatic Compounds, 42 (2020), 1–12. https://doi.org/10.1080/10406638.2020.1748666 doi: 10.1080/10406638.2020.1748666
[25]	C. Liu, Y. H. Pan, J. P. Li, On the Laplacian spectrum and Kirchhoff index of generalized phenylenes, Polycyclic Aromatic Compounds, 41 (2019), 1–10. https://doi.org/10.1080/10406638.2019.1703765 doi: 10.1080/10406638.2019.1703765
[26]	I. Gutman, B. Mohar, The quasi-Wiener and the Kirchhoff indices coincide, J. Chem. Inf. Model., 36 (1996), 982–985.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)