quantitative structure - activity relationships. Curr. Comput. Aided. Drug. Des . 2013 , 9 , 153-163. 5. Gutman, I.; Trinajstić, N. Graph theory and molecular orbitals, Total electron energy of alternant hydrocarbons. Chem. Phys. Lett. 1972 , 17 , 535-538. 6. Gutman, I.; Ruščić, B.; Trinajstić, N; Wilcox, C. F. Graph theory and molecular orbitals. XII. Acyclic polyenes. J. Chem. Phys . 1975 , 62 , 3399-3405. 7. Borovićanin, B.; Das, K. C.; Furtula, B.; Gutman, I. Bounds for Zagreb indices. MATCH Commun. Math. Comput. Chem . 2017 , 78 , 17

### Saba Manzoor, Nisar Fatima, Akhlaq Ahmad Bhatti and Akbar Ali

### Bo Zhao, Jianhou Gan and Hualong Wu

properties. Such indices can be regarded as score functions which map each molecular graph to a non-negative real number. There were many famous degree-based or distance-based indices such as Wiener index, PI index, Zagreb index, atom-bond connectivity index, Szeged index and eccentric connectivity index et al. Because of its wide engineering applications, many works contributed to determining the indices of special molecular graphs (See Yan et al., [ 24 ], Gao and Shi [ 7 ], Gao and Wang [ 8 ], [ 9 ] and [ 10 ] and Jamil et al. [ 13 ] for more details). In our article

### Muhammad Kamran Jamil, Mohammad Reza Farahani, Muhammad Imran and Mehar Ali Malik

Zagreb index M 1 ( G ) and second Zagreb index M 2 ( G ). These are defined as: M 1 ( G ) = Σ ν ∈ V ( G ) d ( ν ) 2 M 2 ( G ) = Σ u ν ∈ V ( G ) d ( u ) ⋅ d ( ν ) $$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {{M_1}(G) = {\Sigma _{\nu \in V(G)}}d{{(\nu )}^2}} \hfill \\ {{M_2}(G) = {\Sigma _{u\nu \in V(G)}}d(u) \cdot d(\nu )} \hfill \\ \end{array} \end{array}$$ In 2010, Todeschini et. al. [ 2 ] and [ 3 ] introduced the multiplicative versions of the above Zagreb indices named as multiplicative Zagreb indices, which are defined as follows

### Prosanta Sarkar, Nilanjan De and Anita Pal

indices of graphene, J. Nanomaterials 2015 , ID 969348. 13. Shigehalli, V.S.; Kanabur, R. Computation of new degree based topological indices of graphene. J. Math. 2016 , ID 4341919. 14. Akhtar, S.; Imran, M.; Gao, W. and Farahani, M.R. On topological indices of honeycomb networks and graphene networks. Hacet. J. Math. Stat. 2018 , 47(1) , 19-35. 15. Kumar, R.P.; Nandappa, D.S. and Kanna, M.R.R., Redefined Zagreb, Randi´c, Harmonic, GA indices of graphene. Int. J. Math. Anal. 2017 , 11(10) , 493-502. 16. Baig, A. Q.; Imran, M

### V.R. Kulli, B. Chaluvaraju and H.S. Boregowda

Zagreb indices , J. Int. Math. Virtual Inst. 7 (2017) 53–67. doi:10.7251/JIMVI1701053G [5] V.R. Kulli, College Graph Theory (Vishwa International Publications, Gulbarga, India, 2012). [6] V.R. Kulli, On K Banhatti indices of graphs , J. Comput. Math. Sci. 7 (2016) 213–218. [7] V.R. Kulli, On K indices of graphs , Internat. J. Fuzzy Math. Arch. 10 (2016) 105–109. [8] V.R. Kulli, The first and second Ka indices and coindices of graphs , Internat. J. Math. Arch. 7 (2016) 71–77. [9] V.R. Kulli, B. Chaluvaraju and H.S. Boregowda, On

### Tihana Škrinjarić

Management of Investment Funds. Harvard Business Review , January-February 1965, pp. 63-75. 30. Wang, P., Moore, T. (2008). Stock market integration for the transition economies: time-varying conditional correlation approach. The Manchester School , Supplement 2008, pp. 116-133. 31. Zagreb Stock Exchange (2015). Trading data and statistics, Indices . Available at http://www.zse.hr [15 October 2015].

### B. Basavanagoud, Wei Gao, Shreekant Patil, Veena R. Desai, Keerthi G. Mirajkar and B Pooja

referred to topological indices [ 16 , 30 ]. Topological indices are found to be very useful in chemistry, biochemistry and nanotechnology in isomer discrimination, structure-property relationship, structure-activity relationship and pharmaceutical drug design. The first and second Zagreb indices of a graph are among the most studied vertex degree based topological indices. The first and second Zagreb indices, respectively defined by M 1 ( G ) = ∑ u ∈ V ( G ) d G ( u ) 2 = ∑ u ν ∈ E ( G ) [ d G ( u ) + d G ( ν ) ] and M 2 ( G ) = ∑ u ν ∈ E ( G ) d G ( u ) d G ( ν

### Hosam Abdo and Darko Dimitrov

. Chem. 66 (2011) 613-626. [16] G.H. Fath-Tabar, Old and new Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 65 (2011) 79-84. [17] I. Gutman and K.C. Das, The first Zagreb index 30 years after , MATCH Commun. Math. Comput. Chem. 50 (2004) 83-92. [18] I. Gutman, P. Hansen and H. M´elot, Variable neighborhood search for extremal graphs. 10. Comparison of irregularity indices for chemical trees, J. Chem. Inf. Model. 45 (2005) 222-230. doi:10.1021/ci0342775 [19] R. Hammack, W. Imrich and S

### B. Basavanagoud, Veena R. Desai and Shreekant Patil

represented by vertices and chemical bonds by edges of a graph. A graphical invariant is a number related to a graph. In other words, it is a fixed number under graph automorphisms. In chemical graph theory, these invariants are also called the topological indices. The first and second Zagreb indices are defined as M 1 ( G ) = Σ u ∈ V ( G ) d G ( u ) 2 = Σ u v ∈ E ( G ) [ d G ( u ) + d G ( v ) ] and M 2 ( G ) = Σ u ∈ V ( G ) d G ( u ) d G ( v ) $$\begin{array}{l} \displaystyle {M_1}(G) = \mathop {\rm{\Sigma }}\limits_{u \in V(G)} {d_G}{(u)^2} = \mathop {\rm

### Najmeh Soleimani, Shila Banari Bahnamiri and Mohammad Javad Nikmehr

-140. 14. Ghorbani, M.; Jalali, M. A simple algorithm for computing topological indices of dendrimers. Iranian Journal of Mathematical Sciences and Informatics . 2007 , 2(2) , 17-23. 15. Buckley, F.; Harary, F. Distance in Graphs. Addison-Wesley, Reading . 1990 . 16. Todeschini, R.; Consonni, V. Handbook of Molecular Descriptors. Weinheim , Wiley-VCH . 2000 . 17. Gutman, I.; Das, K. C. Some properties of the second zagreb index. MATCH Commun. Math. Comput. Chem. 2004 , 50 , 103-112. 18. Ghorbani, M.; Hemmasi, M. Eccentric connectivity