Distance and eccentricity based polynomials and indices of m-level Wheel graph





Hosoya polynomial, Harary polynomial, Schultz polynomial, Eccentric connectivity polynomial, Wiener index, Schultz index, Eccentric connectivity index, m-level wheel graph


Distance and degree based topological polynomial and indices of molecular graphs have various applications in chemistry, computer networking and pharmacy. In this paper, we give hosoya polynomial, Harary polynomial, Schultz polynomial, modified Schultz polynomial, eccentric connectivity polynomial, modified Wiener index, modified hyper Wiener index, generalized Harary index, multiplicative Wiener index, Schultz index, modified Schultz index, eccentric connectivity index of generalized wheel networks Wn,m. We also give pictorial representation of computed topological polynomials and indices on the involved parameters m and n.

Author Biographies

Murat Cancan, Van Yznc Yil University.

Faculty of Education.

Muhammad Hussain, COMSATS University Islamabad.

Dept. of Mathematics.

Haseeb Ahmad, Lahore Leads University.

Dept. of Mathematics.


S. Ahmad, H. M. A. Siddiqui, A. Ali, M. R. Farahani, M. Imran, and I. N. Cangul, “On Wiener index and Wiener polarity index of some polyomino chains”, Journal of discrete mathematical sciences and cryptography, vol. 22, no. 7, pp. 1151–1164, Oct. 2019, doi: 10.1080/09720529.2019.1688965

A. Ali, W. Nazeer, M. Munir, and S. M. Kang, “M-polynomials and topological indices of zigzag and rhombic benzenoid systems”, Open chemistry, vol. 16, no. 1, pp. 73–78, Feb. 2018, doi: 10.1515/chem-2018-0010

U. Ali, Y. Ahmad, and M. S. Sardar, “On 3-total edge product cordial labeling of tadpole, book and flower graphs”, Open Journal of Mathematical Sciences, vol. 4, no. 1, pp. 48–55, Mar. 2020, doi: 10.30538/oms2020.0093

Y. Alizadeh, A. Iranmanesh and S. Mirzaie, ”Computing Schultz Polynomial, Schultz Index of C60 Fullerene By Gap Program”, Digest journal of nanomaterials and biostructures, vol. 4, no. 1, pp. 7-10, 2009. [On line]. Available: https://bit.ly/3hRts8L

A. R. Ashrafi, M. Ghorbani, and M. Jalali, “Eccentric connectivity polynomial of an infinite family of fullerenes”, Optoelectronics and advanced materials – rapid communications, vol. 3, no. 8, pp. 823-826, Jul. 2009. [On line]. Available: https://bit.ly/3doNGTP

F. Asif, Z. Zahid and S. Zafar, ”Leap Zagreb and leap hyperZagreb indices of Jahangir and Jahangir derived graphs”, Engineering and Applied Science Letter, vol. 3, no. 2, pp. 1-8, 2020. [On line]. Available: https://bit.ly/2Bmb1YP

A. Aytac and T. Turaci, “Vertex vulnerability parameter of gear graphs”, International journal of Foundations of Computer Science, vol. 22, no. 05, pp. 1187–1195, Aug. 2011, doi: 10.1142/S0129054111008635

M. Cancan, S. Ediz, and M. R. Farahani, “On ve-degree atom-bond connectivity, sum-connectivity, geometric-arithmetic and harmonic indices of copper oxide”, Eurasian Chemical Communications, vol. 2, no. 5, pp. 641–645, Mar. 2020, doi: 10.33945/SAMI/ECC.2020.5.11

G. G. Cash, “Relationship between the Hosoya polynomial and the hyper-Wiener index”, Applied mathematics letters, vol. 15, no. 7, pp. 893–895, 2002, doi: 10.1016/S0893-9659(02)00059-9

M. V. Diudea. ”Hosoya polynomial in tori”, MATCH communication in mathematical and in computer chemistry, vol. 45, pp. 109-122, 2002. [On line]. Available: https://bit.ly/3ehNKpr

A. A. Dobrynin, R. Entringer, and I. Gutman, “Wiener index of trees: theory and applications”, Acta applicandae mathematicae, vol. 66, no. 3, pp. 211–249, 2001, doi: 10.1023/A:1010767517079

W. Gao and M. R. Farahani, “The hyper-zagreb index for an infinite family of nanostar dendrimer”, Journal of discrete mathematical sciences and cryptography, vol. 20, no. 2, pp. 515–523, Feb. 2017, doi: 10.1080/09720529.2016.1220088

W. Gao and M. R. Farahani, “The Zagreb topological indices for a type of Benzenoid systems jagged-rectangle”, Journal of interdisciplinary mathematics, vol. 20, no. 5, pp. 1341–1348, Jul. 2017, doi: 10.1080/09720502.2016.1232037

W. Gao, L. Shi, and M. R. Farahani, “Szeged related indices of TUAC6[p, q]”, Journal of discrete mathematical sciences and cryptography, vol. 20, no. 2, pp. 553–563, Feb. 2017, doi: 10.1080/09720529.2016.1228312

W. Gao, M. Younas, A. Farooq, A. Virk, and W. Nazeer, “Some reverse degree-based topological indices and polynomials of dendrimers”, Mathematics, vol. 6, no. 10, Art ID. 214, Oct. 2018, doi: 10.3390/math6100214

I. Gutman, S. Klavzar, M. Petkovsekand and P. Zigert. ”On Hosoya polynomials of benzenoid graphs”, MATCH communication in mathematical and in computer chemistry, vol. 43, pp. 49-66, 2001. [On line]. Available: https://bit.ly/312kj74

H. Hosoya, “On some counting polynomials in chemistry”, Discrete applied mathematics, vol. 19, no. 1-3, pp. 239–257, 1988, doi: 10.1016/0166-218X(88)90017-0

S. Klavžar and I. Gutman, “A comparison of the Schultz molecular topological index with the Wiener Index”, Journal of chemical information and computer sciences, vol. 36, no. 5, pp. 1001–1003, Jan. 1996, doi: 10.1021/ci9603689

J.-B. Liu, M. Younas, M. Habib, M. Yousaf, and W. Nazeer, “M-polynomials and degree-based topological indices of VC5C7[p,q] and HC5C7[p,q] nanotubes”, IEEE access, vol. 7, pp. 41125–41132, 2019, doi: 10.1109/ACCESS.2019.2907667

B. Mohar and T. Pisanski, “How to compute the Wiener index of a graph”, Journal of mathematical chemistry, vol. 2, no. 3, pp. 267–277, Jun. 1988, doi: 10.1007/BF01167206

M. Munir, W. Nazeer, S. Rafique, and S. Kang, “M-polynomial and degree-based topological indices of polyhex nanotubes”, Symmetry, vol. 8, no. 12, Art ID. 149, Dec. 2016, doi: 10.3390/sym8120149

H. P. Schultz, “Topological organic chemistry. 1. Graph theory and topological indices of alkanes”, Journal of chemical information and modeling, vol. 29, no. 3, pp. 227–228, Aug. 1989, doi: 10.1021/ci00063a012

A. Shah and S. A. U. H. Bokhary, “On chromatic polynomial of certain families of dendrimer graphs”, Open journal of mathematical sciences, vol. 3, no. 1, pp. 404–416, Dec. 2019, doi: 10.30538/oms2019.0083

V. Sharma, R. Goswami, and A. K. Madan, “Eccentric connectivity index: A novel highly discriminating topological descriptor for structure−property and structure−activity studies”, Journal of chemical information and computer sciences, vol. 37, no. 2, pp. 273–282, 1997, doi: 10.1021/ci960049h

H. M. A. Siddiqui and M. Imran, “Computing the metric dimension of wheel related graphs”, Applied mathematics and computation, vol. 242, pp. 624–632, Sep. 2014, doi: 10.1016/j.amc.2014.06.006

D. Stevanović, “Hosoya polynomial of composite graphs”, Discrete mathematics, vol. 235, no. 1-3, pp. 237–244, May 2001, doi: 10.1016/S0012-365X(00)00277-6

A. Tabassum, M. A. Umar, M. Perveen, and A. Raheem, “Antimagicness of subdivided fans”, Open Journal of Mathematical Sciences, vol. 4, no. 1, pp. 18–22, Feb. 2020, doi: 10.30538/oms2020.0089

I. Tomescu, I. Javaid and I. Slamin, ”On the partition dimension and connected partition dimension of wheels”, Ars combinatoria, vol. 84, pp.311-318, 2007.

T. Turaci, “The average lower 2-domination number of wheels related graphs and an algorithm”, Mathematical and computational applications, vol. 21, Art ID. 29, Jul. 2016, doi: 10.20944/preprints201607.0037.v1

M. A. Umar, N. Ali, A. Tabassum and B. R. Ali, ”Book graphs are cycle antimagic”, Open Journal of Mathematical Sciences, vol. 3, no. 1, pp. 184-90, 2019, Jun. doi: 10.30538/oms2019.0061

H. Wiener, “Structural determination of paraffin boiling points”, Journal of the American Chemical Society, vol. 69, no. 1, pp. 17–20, Jan. 1947, doi: 10.1021/ja01193a005

L. Yang, “Wiener index and traceable graphs”, Bulletin of the Australian Mathematical Society, vol. 88, no. 3, pp. 380–383, Dec. 2013, doi: 10.1017/S0004972712000901



How to Cite

M. Cancan, M. Hussain, and H. Ahmad, “Distance and eccentricity based polynomials and indices of m-level Wheel graph”, Proyecciones (Antofagasta, On line), vol. 39, no. 4, pp. 869-885, Jul. 2020.

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