Generalized Zagreb index of some dendrimer structures

Year 2018, Volume: 1 Issue: 3, 160 - 165, 30.09.2018

Prosanta Sarkar Nilanjan De , İsmail Naci Cangül , Anita Pal

https://doi.org/10.32323/ujma.425103

Cited By: 1

Abstract

Chemical graph theory, is a branch of mathematical chemistry which deals with the nontrivial applications of graph theory to solve molecular problem. A chemical graph is represent a molecule by considering the atoms as the vertices and bonds between them as the edges. A topological index is a graph based molecular descriptor, which is graph theoretic invariant characterising some physicochemical properties of chemical compounds. Dendrimers are generally large, complex, and hyper branched molecules synthesized by repeatable steps with nanometre scale measurements. In this paper, we study the $(a,b)$-Zagreb index of some regular dendrimers and hence obtain some vertex degree based topological indices.

Keywords

Dendrimers, Generalized Zagreb index, Vertex degree-based topological indices

References

• [1] D.A. Tomalia, H. Baker, J.R. Dewald, M. Hall, G. Kallos, S. Martin, J. Roeek, J. Ryder, and P. Smith, A new class of polynomials: starburst-dendriticmacromolecules, Polym. J., 17, (1985), 117-132.
• [2] U. Ahmad, S. Ahmad, and R. Yousaf, Computation of Zagreb and atom–bond connectivity indices of certain families of dendrimers by usingautomorphism group action, J. Serb. Chem. Soc., 82, (2),(2017), 151-162.
• [3] Y. Bashir, A. Aslam, M. Kamran, M.I. Qureshi, A. Jahangir, M. Rafiq, N. Bibi, and N. Muhammad, On forgotten topological indices of some dendrimersstructure, Molecules, 22, (867), (2017), 1-8.
• [4] I. Gutman, N. Trinajesti´c, Graph theory and molecular orbitals total -electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17, (1972),535-538.
• [5] N. De, The vertex Zagreb indices of some graphs operations, Carpathian Math. Publ., 8, (2), (2016), 215-223.
• [6] P. Sarkar, N. De, and A. Pal, The Zagreb indices of graphs based on new operations related to the join of graphs, J. Int. Math. Virtual Inst., 7,(2017),181-209.
• [7] P. Sarkar, N. De., and A. Pal, F-index of graphs based on new operations related to the join of graphs, arXiv:1709.06301v1.
• [8] N. De, On molecular topological properties of TiO2 nanotubes, J. Nanoscience, 2016, (2016), DOI: 1028031.
• [9] S. Akhtar, M. Imran, Computing the forgotten topological index of four operations on graphs, AKCE Int. J. Graphs Comb., 14, (1), (2017), 70-79.
• [10] P.S. Ranjini, V. Lokesha, and A. Usha, Relation between phenylene and hexagonal squeeze using harmonic index, Int. J. Graph Theory, 1,(2013),116-121.
• [11] B. Zhao, J. Gan, and H. Wu, Redefined Zagreb indices of some nano structures, Appl. Math. Nonlinear Sci., 1, (1), (2016), 291-300.
• [12] R.P. Kumar, D.S. Nandappa, and M.R.R. Kanna, Redefined zagreb, Randi´c, Harmonic, GA indices of graphene, Int. J. Math. Anal., 11, (10), (2017),493-502.
• [13] X. Li, J. Zheng, A unified approach to the extremal trees for dierent indices, MATCH Commun. Math. Comput. Chem., 54, (2005), 195-208.
• [14] I. Gutman, M. Lepovi´c, Choosing the exponent in the definition of the connectivity index, J. Serb. Chem. Soc., 66,(9),(2001), 605-611.
• [15] V. Lokesha, T. Deepika, Symmetric division deg index of tricyclic and tetracyclic graphs, Int. J. Sci. Eng. Res, 7, (5), (2016), 53-55.
• [16] V. Alexander, Upper and lower bounds of symmetric division deg index, Iran. J. Math. Chem., 52, (2014), 91-98.
• [17] C.K. Gupta, V. Lokesha, B.S. Shwetha, and P.S. Ranjini, Graph operations on the symmetric division deg index of graphs, Palestine. J. Math., 6, (1),(2017), 280-286.
• [18] M. Azari, A. Iranmanesh, Generalized Zagreb index of graphs, Studia Univ. Babes-Bolyai., 56,(3), (2011), 59-70.
• [19] M. R. Farahani, The generalized Zagreb index of circumcoronene series of benzenoid, J. Appl. Phys. Sci. Int., 3,(3), (2015), 99-105.
• [20] P. Sarkar, N. De, and A. Pal, The (a,b)-Zagreb index of nanostar dendrimers, preprint.[21] M. R. Farahani, M. R. R. Kanna, Generalized Zagreb index of V-phenylenic nanotubes and nanotori, J. Chem. Pharm. Res., 7,(11), (2015), 241-245.