Laplacian and signless laplacian spectra and energies of multi-step wheels

Zheng-Qing Chu; Mobeen Munir; Amina Yousaf; Muhammad Imran Qureshi; Jia-Bao Liu; Zheng-Qing Chu; Mobeen Munir; Amina Yousaf; Muhammad Imran Qureshi; Jia-Bao Liu

doi:10.3934/mbe.2020206

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 4: 3649-3659. doi: 10.3934/mbe.2020206

Previous Article Next Article

Research article Special Issues

Laplacian and signless laplacian spectra and energies of multi-step wheels

1.
Department of Mathematics and Physics, Anhui Xinhua University, Hefei 230088, China
2.
Division of Science and Technology, University of Education, Lahore 54000, Pakistan
3.
Department of Mathematics, COMSATS University Islamabad, Vehari campus, Vehari 61100, Pakistan
4.
School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China

Received: 25 January 2020 Accepted: 06 May 2020 Published: 19 May 2020

Energies and spectrum of graphs associated to different linear operators play a significant role in molecular chemistry, polymerisation, pharmacy, computer networking and communication systems. In current article, we compute closed forms of signless Laplacian and Laplacian spectra and energies of multi-step wheel networks W_{n, m}. These wheel networks are useful in networking and communication, as every node is one hoop neighbour to other. We also present our results for wheel graphs as particular cases. In the end, correlation of these energies on the involved parameters m ≥ 3 and n is given graphically. Present results are the natural generalizations of the already available results in the literature.
- laplacian matrix,
- signless laplacian matrix,
- spectrum,
- laplacian energy,
- signless laplacian energy,
- wheel graphs
Citation: Zheng-Qing Chu, Mobeen Munir, Amina Yousaf, Muhammad Imran Qureshi, Jia-Bao Liu. Laplacian and signless laplacian spectra and energies of multi-step wheels[J]. Mathematical Biosciences and Engineering, 2020, 17(4): 3649-3659. doi: 10.3934/mbe.2020206

Related Papers:

Abstract

Energies and spectrum of graphs associated to different linear operators play a significant role in molecular chemistry, polymerisation, pharmacy, computer networking and communication systems. In current article, we compute closed forms of signless Laplacian and Laplacian spectra and energies of multi-step wheel networks W_{n, m}. These wheel networks are useful in networking and communication, as every node is one hoop neighbour to other. We also present our results for wheel graphs as particular cases. In the end, correlation of these energies on the involved parameters m ≥ 3 and n is given graphically. Present results are the natural generalizations of the already available results in the literature.

References

[1]	S. Wang, A. Gittens, M. W. Mahoney, Scalable kernel K-Means clustering with nystrom approximation: Relative-Error bounds, J. Mach. Learn. Res., 20 (2019), 1-49.
[2]	E. A. Castro, G. Chen, G. Lerman, Spectral clustering based on local linear approximations, Electron. J. Stat., 5 (2011), 1537-1587.
[3]	P. Daugulis, A note on a generalization of eigenvector centrality for bipartite graphs and applications, Networks, 59 (2012), 261-264. doi: 10.1002/net.20442
[4]	D. J. Griffiths, Introduction to Quantum Mechanics (2^nd edition), Prentice Hall, (2004).
[5]	F. Laloe, Do We Really Understand Quantum Mechanics (2^nd edition), Cambridge University Press, (2019).
[6]	M. V. Diudea, I. Gutman, J. Lorentz, Molecular Topology, Nova Science Publishers, (2001).
[7]	G. Bieri, J. D. Dill, E. Heilbronner, A. Schmelzer, Application of the equivalent bond orbital model to the C2s-Ionization energies of saturated hydrocarbons, Helv. Chim. Acta., 60 (1977), 2234-2247. doi: 10.1002/hlca.19770600715
[8]	E. Heilbronner, A simple equivalent bond orbital model for the rationalization of the C2s-Photoelectron spectra of the higher n-Alkanes, in particular of polyethylene, Helv. Chim. Acta., 60 (1977), 2248-2257. doi: 10.1002/hlca.19770600716
[9]	H. Gunthard, H. Primas, Zusammenhang von Graphentheorie und MO-Theorie von Molekeln mit systemen konjugierter bindungen, Helv. Chim. Acta., 39 (1956), 1645-1653. doi: 10.1002/hlca.19560390623
[10]	S. Meenakshi, S. Lavanya, A survey on energy of graphs, Ann. Pure Appl. Math., 8 (2014), 183-191.
[11]	G. Indulal, A. Vijaykumar, Energies of some non-regular graphs, J. Math. chem., 42 (2007), 377-386. doi: 10.1007/s10910-006-9108-7
[12]	M. Jooyandeh, D. Kiani, M. Mirzakhan, Incidence energy of a graph, MATCH. Commun. Math. Comput. Chem., 62 (2009), 561-572.
[13]	I. Gutman, The energy of a graph, Ber. Math. Stat. Sekt. Forschungszent. Graz, 103 (1978), 1-22.
[14]	I. Gutman, B. Zhou, Laplacian energy of a graph, Lin. Algebra Appl., 414 (2006), 29-37. doi: 10.1016/j.laa.2005.09.008
[15]	B. Zhou, I. Gutman, On Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem., 57 (2007), 211-220.
[16]	I. Gutman, G. Indulal, A. Vijaykumar, On distance energy of graphs, MATCH Commun. Math. Compul. Chem., 60 (2008), 461-472.
[17]	I. Gutman, X. Li, Y. Shi, Graph Energy, Springer, (2012).
[18]	V. Nikiforov, The energy of graphs and matrices, Jour. Math. Anal. Appl., 326 (2007), 1472-1475. doi: 10.1016/j.jmaa.2006.03.072
[19]	D. Cvethovic, P. Rowlinson, K. Simic, Signless Laplacians of finite graphs, MATCH Commun. Math. Comput. Chem., 57 (2007), 211-220.
[20]	T. Turaci, The average lower 2-domination number of wheels related graphs and an algorithm, Math. Comput. Appl., 21 (2016), 1-9.
[21]	A. Aytac and T. Turaci, Vertex vulnerablility parameter of Gear Graphs, Int. J. Found. Comput. Sci., 22 (2011), 1187-1195. doi: 10.1142/S0129054111008635
[22]	J. B. Liu, M. Munir, A. Yousaf, A. Naseem, K. Ayub, Distance and Adjacency energies of MultiLevel wheel networks, Mathematics, 7 (2019), 1-9.
[23]	J. B. Liu, X. F. Pan, F. T. Hu, Asymptotic Laplacian-energy-like invariant of lattices, Appl. Math. Comput., 253 (2015), 205-214.
[24]	J. B. Liu, X. F. Pan, Asymptotic incidence energy of lattices, Phy. A, 422 (2015), 193-202. doi: 10.1016/j.physa.2014.12.006
[25]	I. Tomescu, I. Javaid, Slamin, On the partition dimension and connected partition dimension of wheels, Ars Comb., 84 (2007), 311-317.
[26]	H. M. A. Siddique, H. Imran, Computing the metric dimension of wheel related graphs, Appl. Math. Comput., 242 (2014), 624-632.
[27]	Z. Hussain, S. M. Kang, M. Rafique, M. Munir, U. ALi, A. Zahid, et al., Bounds for partition dimension of m-Wheels, Open Phy., 17 (2019), 340-344. doi: 10.1515/phys-2019-0037

Reader Comments

Your name:*

Email:*
© 2020 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)