M-polynomial and related degree-based topological indices of the third type of chain Hex-derived network

Shibsankar Das; Shikha Rai

doi:10.26637/MJM0804/0085

Abstract

In chemical graph theory, a topological index is a numerical descriptor that describes the various biological activities, physical properties and chemical reactivities of molecular graphs. Recent studies compute several
degree-based topological indices of a graph network by deriving its M-polynomial. In this paper, we would like to find out a closed form of M-polynomial for the third type of chain Hex-derived network of dimension n and hence derive various degree-based topological indices. Further, we plot the M-polynomial and all the related degree-based topological indices of the above network for different dimensional values.

Keywords:

Degree-based topological indices, M-polynomial,, Chain Hex-derived network, Graph polynomial

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Shibsankar Das Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, Uttar Pradesh, India. https://orcid.org/0000-0003-0082-6673
Shikha Rai Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, Uttar Pradesh, India.

Pages: 1842-1850
Date Published: 01-01-2020
Vol. 8 No. 04 (2020): Malaya Journal of Matematik (MJM)

D. B. West, Introduction to Graph Theory, 2nd Edition, Prentice Hall, 2000.

R. Hammack, W. Imrich, S. Klavžar, Handbook of Product Graphs, 2nd Edition, CRC Press, Inc., Boca Raton, FL, USA, 2011.

E. Estrada, Randić index, irregularity and complex biomolecular networks, Acta ChimicaSlovenica 57 (2010) $597-603$.

R. García-Domenech, J. Gálvez, J. V. de Julián-Ortiz, L. Pogliani, Some new trends in chemical graph theory, Chemical Reviews 108 (3) (2008) 1127-1169.

A. T. Balaban, Chemical applications of graph theory, Mathematical Chemistry, Academic Press, 1976.

N. Trinajstić, Chemical Graph Theory, 2nd Edition, Mathematical Chemistry Series, CRC Press, 1992.

J. L. Gross, J. Yellen, P. Zhang, Handbook of Graph Theory, 2nd Edition, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, 2013.

I. Gutman, Degree-based topological indices, CroaticaChemica Acta 86 (4) (2013) 351-361.

A. T. Balaban, Highly discriminating distance-based topological index, Chemical Physics Letters 89 (5) (1982) 399-404.

K. Pattabiraman, Degree and distance based topological indices of graphs, Electronic Notes in Discrete Mathematics 63 (2017) 145-159.

P. V. Khadikar, N. V. Deshpande, P. P. Kale, A. Dobrynin, I. Gutman, G. Domotor, The szeged index and an analogy with the wiener index, Journal of Chemical Information and Computer Sciences 35 (3) (1995) 547-550.

I. Gutman, The acyclic polynomial of a graph, Publications de I'InstitutMathématique 22(36) (42) (1977) 63-69.

E. J. Farrell, An introduction to matching polynomials, Journal of Combinatorial Theory, Series B 27 (1) (1979) $75-86$.

H. Zhang, F. Zhang, The clar covering polynomial of hexagonal systems I, Discrete Applied Mathematics 69 (1-2) (1996) 147-167.

I. Gutman, Some relations between distance-based polynomials of trees, Bulletin (Académie serbe des sciences et des arts. Classe des sciences mathématiques et naturelles. Sciences mathématiques) 131 (30) (2005) 1-7.

L. H. Kauffman, A tutte polynomial for signed graphs, Discrete Applied Mathematics 25 (1-2) (1989) 105-127.

H. Hosoya, On some counting polynomials in chemistry, Discrete Applied Mathematics 19 (1-3) (1988) 239-257.

H. Wiener, Structural determination of paraffin boiling points, Journal of the American Chemical Society 69 (1) (1947) 17-20.

E. Deutsch, S. Klavžar, M-polynomial and degree-based topological indices, Iranian Journal of Mathematical Chemistry 6 (2) (2015) 93-102.

M. Munir, W. Nazeer, S. Rafique, S. M. Kang, Mpolynomial and degree-based topological indices of polyhex nanotubes, Symmetry 8 (12) (2016) 149.

M. Munir, W. Nazeer, S. Rafique, S. M. Kang, Mpolynomial and related topological indices of nanostar dendrimers, Symmetry 8 (9) (2016) 97.

M. Munir, W. Nazeer, A. R. Nizami, S. Rafique, S. M. Kang, M-polynomial and topological indices of titania nanotubes, Symmetry 8 (1-9) (2016) 117.

Y. C. Kwun, M. Munir, W. Nazeer, S. Rafique, S. M.Kang, M-polynomials and topological indices of $mathrm{V}$ phenylenic nanotubes and nanotori, Scientific Reports 7 (1) (2017) 1-9.

S. M. Kang, W. Nazeer, M. A. Zahid, A. R. Nizami, A. Aslam, M. Munir, M-polynomials and topological indices of hex-derived networks, Open Physics 16 (1) (2018) 394-403.

H. Deng, J. Yang, F. Xia, A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes, Computers & Mathematics with Applications 61 (10) (2011) 3017-3023.

I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. total $pi$-electron energy of alternant hydrocarbons. Chemical Physics Letters 17 (4) (1972) 535-538.

A. Miličević, S. Nikolić, N. Trinajstić, On reformulated zagreb indices, Molecular Diversity 8 (2004) 393-399.

M. Randić, Characterization of molecular branching, Journal of the American Chemical Society 97 (23) (1975) 6609-6615.

B. Bollobás, P. Erdős, Graphs of extremal weights, Ars Combinatoria 50 (1998) 225-233.

D. Amić, D. Bešlo, B. Lučić, S. Nikolić, N. Trinajstić, The vertex-connectivity index revisited, Journal of Chemical Information and Computer Sciences 38 (5) (1998) 819-822.

D. Vukičević, M. Gašperov, Bond additive modeling 1. adriatic indices, CroaticaChemica Acta 83 (3) (2010) 243-260.

J. Sedlar, D. Stevanović, A. Vasilyev, On the inverse sum indeg index, Discrete Applied Mathematics 184 (2015) 202-212.

B. Furtula, A. Graovac, D. Vukičević, Augmented zagreb index, Journal of Mathematical Chemistry 48 (2) (2010) 370-380.

F. G. Nocetti, I. Stojmenovic, J. Zhang, Addressing and routing in hexagonal networks with applications for tracking mobile users and connection rerouting in cellular networks, IEEE Transactions on Parallel and Distributed Systems 13 (9) (2002) 963-971.

P. Manuel, R. Bharati, I. Rajasingh, C. Monica M, On minimum metric dimension of honeycomb networks, Journal of Discrete Algorithms 6 (1) (2008) 20-27.

F. S. Raj, A. George, On the metric dimension of HDN 3 and PHDN 3, in: 2017 IEEE International Conference on Power, Control, Signals and Instrumentation Engineering (ICPCSI), 2017, pp. 1333-1336.

O. Favaron, M. Mahéo, J.-F. Saclé, Some eigenvalue properties in graphs (conjectures of graffiti-II), Discrete Mathematics 111 (1-3) (1993) 197-220.

S. Hayat, M. Imran, Computation of topological indices of certain networks, Applied Mathematics and Computation 240 (2014) 213-228.

F. S. Raj, A. George, Network embedding on planar octahedron networks, in: 2015 IEEE International Conference on Electrical, Computer and Communication Technolo-gies (ICECCT), IEEE, 2015, pp. 1-6.

C.-C. Wei, H. Ali, M. A. Binyamin, M. N. Naeem, J.-B. Liu, Computing degree based topological properties of third type of hex-derived networks, Mathematics 7 (4) (2019) 368.

S. Das, S. Rai, M-polynomial and related degree-based topological indices of the third type of hex-derived network, Nanosystems: Physics, Chemistry, Mathematics 11 (3) (2020) 267-274.