Document Type : Review Article


1 Department of Mathematics and Statistics, Institute of Southern Punjab, Multan, Pakistan

2 Department of Mathematics, The University of Lahore, Pakpattan Campus, Pakistan

3 School of Information Science and Technology, Yunnan Normal University, Kunming, China

4 School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, P.R. China



Graph theory plays a vital role in modeling and designing any chemical structure or chemical network. Chemical graph theory helps to understand the molecular structural properties of a molecular graph. The molecular graph is a graph consisting of atoms called vertices and the chemical bond between atoms called edges. In this paper, we have computed the various types of degree-based fifth M-Zagreb indices, general fifth M-Zagreb indices, fifth hyper-M-Zagreb indices, general fifth M1-Zagreb polynomial, general fifth M2-Zagreb polynomial, fifth M1-Zagreb polynomial and fifth M2-Zagreb polynomial, fifth hyper-M1-Zagreb polynomial, fifth hyper-M2-Zagreb polynomial  and fifth M3-Zagreb index and their polynomials of a molecular graph namely carbon graphite denoted by CG(m,n) for t-levels.

Graphical Abstract

General fifth M-Zagreb indices and fifth M-Zagreb polynomials of carbon graphite


Main Subjects

[1] A.Q. Baig, M. Imran, W. Khalid, M. Naeem, Can. J. Chem, 2017, 95, 674-686.
[2] M.R. Farahani, J. Math. Nanosci., 2012, 2, 15-20.
[3] W. Gao, W.F. Wang, J. of Diff. Equ. & App., 23, 2016, 1-10.
[4] A. Graovac, M. Ghorbani, M.A. Hosseinzadeh, J. of Math. Nanosci., 2011, 1, 33-42.
[5] M.N. Husin, R. Hasni, N.E. Arif, M. Imran, Molecules, 2016, 21, 821.
[6] M. Imran, M. Naeem, A. Qadair Baig, M. K. Siddiqui, M.A. Zahid, W. Gao, J. of Dis. Math. Sci. and Cry., 2019, 22, 1215-1228.
[7] M. Imran, A.A.E. Abunamous, D. Adi, S.H. Rafique, A.Q. Baig, M.R. Farahani, J. of Dis. Math. Sci. and Cry., 2019, 22, 1199-1213.
[8] V.R. Kulli, Intern. J. Fuzzy Math. Arch. 2017, 13, 99-103.
[9] V.R. Kulli, College Graph Theory, Vishwa International Publications, Gulbarga, India, 2012.
[10] V.R. Kulli, Int. J. Math. Arch., 2017, 8, 103-109.
[11] J.B. Liu, A.Q. Baig, W. Khalid, M.R. Farahani, Com. Ren. de l’Acad. bulg. des Sci., Tom.,  2018, 71, 10-21.
[12] H. Yang, M. Naeem, A.Q. Baig, H. Shaker, M.K. Siddiqui. J. Dis. Math. Sci. and Cryp., 2019, 22, 1177-1187.
[13] M.A. Zahid, M. Naeem, A.Q. Baig, Wei Gao, Utilit. Math., 2018, 109, 263-270.