### Four new operations related to composition and their reformulated Zagreb index

#### Abstract

_{1}(G) of a simple graph G is defined as the sum of the terms (d

_{u}+d

_{v}-2)

^{2}over all edges uv of G. In 2017, Sarala et al. introduced four new operations (F-product) of graphs. In this paper, we study the first reformulated Zagreb index for the F-product of some special well-known graphs such as subdivision and total graph.

#### Keywords

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H. Deng, D. Sarala, S.K. Ayyaswamy, S. Balachandran, The Zagreb indices of four operations on graphs, Appl. Math. Comput. 275 (2016) 422–431.

M. Eliasi, B. Taeri, Four new sums of graphs and their wiener indices, Discrete Appl. Math. 157 (2009) 794–803.

D. Sarala, H. Deng, S.K. Ayyaswamy, S. Balachandran, The Zagreb indices of graphs based on four new operations related to the lexicographic product, Appl. Math. Comput. 309 (2017) 156–169.

J. Devillers, A.T. Balaban, Eds., Topological indices and related descriptors in QSAR and QSPR, Gordon and Breach, Amsterdam, The Netherlands, 1999.

I. Gutman, O.E. Polansky, Mathematical concepts in organic chemistry, Springer-verlag, Berlin 1986.

I. Gutman, N. Trinajstic ́, graph theory and molecular orbits. Total π−election energy of al- ternant hydrocarbons, Chem. Phy. Lett. 17 (1972) 535–538.

M.H.Khalifeh,H.Yousefi-Azari, A.R.Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math. 157 (2009) 804–811.

L. Feng, A. Ilic, Zagreb, Harary and hyper-Wiener indices of graphs with a given matching number, Appl. Math. Lett. 23 (2010) 943–948.

I. Gutman,K.C.Das,The first Zagreb index 30 yearsafter, MATCH Commun.Math.Comput. Chem. 50 (2004) 83–92.

H. Hua, S. Zhang, Relations between Zagreb coindices and some distance-based topological indices, MATCH Commun. Math.Comput. Chem. in press.

G.H. Shirdel. H.Rezapour, A.M. Sayadi, The hyper-Zagreb index of graph operations, Ira- nian. J. Math. Chem. 4 (2013) 213–220.

Gutman, B. Furtula, Z. Kovijanic Vukicevic, G. Popivoda, Zagreb indices and coindices, MATCH Commun. Math. Comput. Chem. 74 (2015) 5–16.

M.R. Farahani, Computing the hyper-Zagreb index of hexagonal nanotubes, J. Chem.& Ma- terials Research 2 (2015) 16–18.

M.R. Farahani, The hyper-Zagreb index of TUSC4C8(S) nanotubes, Int. J. Engg.& Tech. Research, 2 (2015) 16–18.

G.H. Shirdel, H. Rezapour and A.M. Sayadi, The hyper-Zagreb index of graph operations, Iranian J. Math. Chem. 4 (2013) 213–220.

B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015), 1184–1190.

H.Deng, D.Sarala, S.K.Ayyaswamy, S.Balachandran, The Zagreb indices of four operations on graphs, Appl. Math. Comput. 275 (2016) 422–431.

S. Ghobadi, M. Ghorbaninejad, The forgotten topological index of four operations on some special graphs, Bull. Math. Sci. and Appl. 16 (2016) 89–95.

F.Falahati-Nezhad, M.Azari, Bounds on the hyper-Zagreb index, J. Appl. Math.& Informatics 34 (2016) 319 – 330.

K. Pattabiraman, M. Vijayaragavan, Hyper-Zagreb indices and its coindices of graphs, Bull. Int. Math. Virtual Institute 7 (2017) 31–41.

N.De, Some bounds of reformulated Zagreb indices, Appl. Math. Sci. 101 (2012)5005–5012.

A. Ilic ́, B. Zhou, On reformulated Zagreb indices, Discrete Appl. Math. 160 (2012) 204–209.

A. Milic ́evic ́, S.Nikolic ́, N.Trinajstic ́, On reformulated Zagreb indices, Mol. Divers. 8 (2004) 93–399.

G. Su, L. Xiong, L. Xu, B. Ma, On the maximum and minimum first reformulated Zagreb index with connectivity of at most k, Filomat 25 (2011) 75–83.

B. Zhou, N. Trinajstic ́, Some properties of the reformulated Zagreb indices, J. Math. Chem. 48 (2010) 714–719.

DOI: http://dx.doi.org/10.19184/ijc.2018.2.1.5

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