On the Second Order First Zagreb Index

Document Type : Research Paper

Authors

1 KARNATAK UNIVERSITY DHARWAD

2 Karnatak University

3 Key Laboratoryof High Performance Computing and Stochastic Information Processing, College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan, 410081, P. R. China

Abstract

Inspired by the chemical applications of higher-order connectivity index (or Randic index), we consider here the higher-order first Zagreb index of a molecular graph. In this paper, we study the linear regression analysis of the second order first Zagreb index with the entropy and acentric factor of an octane isomers. The linear model, based on the second order first Zagreb index, is better than models corresponding to the first Zagreb index and F-index. Further, we compute the second order first Zagreb index of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8[p; q], tadpole graphs, wheel graphs and ladder graphs.

Keywords

Main Subjects


[1] A. R. Ashrafi, S. Yousefi, Computing Wiener index of a š‘‡š‘ˆš¶ō€¬øš¶ō€¬¼(š‘†) nanotorus,
MATCH Commun. Math. Comput. Chem.57 (2007) 403–410.
[2] R Core Team, R: A language and environment for statistical computing. R
Foundation for Statistical Computing, Vienna, Austria. (2016) URL: https://www.Rproject.
org/.
[3] D. de Caen, An upper bound on the sum of squares of degrees in a graph, Discrete
Math. 185 (1998) 245–248.
[4] K. C. Das, Sharp bounds for the sum of the squares of the degrees of a graph,
Kragujevac J. Math. 25 (2003) 31–49.
[5] N. De, S. M. A. Nayeem, A. Pal, The F-coindex of some graph operations, Springer
Plus (2016) 5:221, doi: 10.1186/s40064-016-1864-7.
[6] H. Deng, Catacondensed Benzenoids and Phenylenes with the extremal third–order
Randić index, MATCH Commun. Math. Comput. Chem.64 (2010) 471– 496.
[7] M. V. Diudea, E. C. Kirby, The energetic stability of tori and single wall tubes,
Fuller. Sci. Technol. 9 (2001) 445–465.
[8] M. V. Diudea, Graphenes from 4-valent tori, Bull. Chem. Soc. Jpn. 75 (2002) 487–
492.
[9] M. V. Diudea, Hosoya polynomial in tori, MATCH Commun. Math. Comput. Chem.
45 (2002) 109–122.
[10] M. V. Diudea, B. Parv, E. C. Kirby, Azulenic tori, MATCH Commun. Math.
Comput. Chem. 47 (2003) 53–70.
[11] M. V. Diudea, M. Stefu, B. Prv, P. E. John, Wiener index of armchair polyhex
nanotubes, Croat. Chem. Acta 77 (12) (2004) 111–115.
[12] B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015)
1184–1190.
[13] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total šœ‹-electron
energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538.
[14] F. Harary, Graph Theory, Addison-Wesley, Reading, Mass., 1969.
[15] S. M. Hosamani, Computing Sanskruti index of certain nanostructures, J. Appl.
Math. Comput. 54 (1−2) (2017) 425–433.
[16] L. B. Kier, L. H. Hall, Molecular Connectivity in Chemistry and Drug Research,
Academic Press, New York, 1976.
[17] L. B. Kier, L. H. Hall, W. J. Murray, M. Randić, Molecular connectivity V:
connectivity series concept applied to density, J. Pharm. Sci. 65 (1976) 1226–1230.
[18] A. Miličević, S. Nikolić, On variable Zagreb indices, Croat. Chem. Acta 77 (1−2)
(2004) 97–101.
[19] A. Miličević, N. Raos, Estimation of stability of coordination compounds by using
topological indices, Polyhedron 25 (14) (2006) 2800–2808.
[20] M. F. Nadeem, S. Zafar, Z. Zahid, On certain topological indices of the line graph of
subdivision graphs, Appl. Math. Comput. 271 (2015) 790–794.
[21] M. F. Nadeem, S. Zafar, Z. Zahid, On topological properties of the line graphs of
subdivision graphs of certain nanostructures, Appl. Math. Comput. 273 (2016) 125–
130.
[22] S. Nikolić, N. Trinajstić, S. Ivaniš, The connectivity indices of regular graphs,
Croat. Chem. Acta 72 (4) (1999) 875–883.
[23] S. Nikolić, N. Raos, Estimation of stablity constants of mixed amino acid complexes
with copper (II) from topological indices, Croat. Chem. Acta 74 (2001) 621–631.
[24] L. Pogliani, Molecular modeling by linear combinations of connectivity indices, J.
Phys. Chem. 99 (3) (1995) 925–937.
[25] L. Pogliani, A strategy for molecular modeling of a physicochemical property using
a linear combination of connectivity indexes, Croat. Chem. Acta 69 (1) (1996) 95–
109.
[26] L. Pogliani, Higher-level descriptor in molecular connectivity, Croat. Chem. Acta 75
(2) (2002) 409–432.
[27] J. Rada, O. Araujo, I. Gutman, Randić index of benzenoid systems and phenylenes,
Croat. Chem. Acta 74 (2001) 225–235.
[28] J. Rada, O. Araujo, Higher order connectivity indices of starlike trees, Discrete Appl.
Math. 119 (2002) 287–295.
[29] P. S. Ranjini, V. Lokesha, M. A. Rajan, On the Shultz index of the subdivision
graphs, Adv. Stud. Contemp. Math. 21 (3) (2011) 279–290.
[30] P. S. Ranjini, V. Lokesha, I. N. Cangül, On the Zagreb indices of the line graphs of
the subdivision graphs, Appl. Math. Comput. 218 (2011) 699–702.
[31] M. Randić, On characterization of molecular branching, J. Am. Chem. Soc. 97
(1975) 6609–6615.
[32] G. Su, L. Xu, Topological indices of the line graph of subdivision graphs and their
Schur-bounds, Appl. Math. Comput. 253 (2015) 395–401.
[33] P. Tattar, S. Ramaiah, B. G. Manjunath, A Course in Statistics with R, John Wiley &
Sons, Ltd., 2016.
[34] E. W. Weisstein, Tadpole graph, From Matheworld-A Wolfram Web Resource.
[35] J. Zhang, H. Deng, S. Chen, Second order Randic′ index of phenylenes and their
corresponding hexagonal squeezes, J. Math. Chem. 42 (4) (2007) 941–947.
[36] J. Zhang, H. Deng, Third order Randić index of phenylenes, J. Math. Chem. 43 (1)
(2008) 12–18.