Bounds for neighborhood Zagreb index and its explicit expressions under some graph operations

Authors

DOI:

https://doi.org/10.22199/issn.0717-6279-2020-04-0050

Keywords:

Topological index, First Zagreb index, Second Zagreb index, Neighbouhood Zagreb index, Graph operations

Abstract

Topological indices are useful in QSAR/QSPR studies for modeling biological and physiochemical properties of molecules. The neighborhood Zagreb index (MN) is a novel topological index having good correlations with some physiochemical properties. For a simple connected graph G, the neighborhood Zagreb index is the totality of square of ?G(v) over the vertex set, where ?G(v) is the total count of degrees of all neighbors of v in G. In this report, some bounds are established for the neighborhood Zagreb index. Some explicit expressions of the index for some graph operations are also computed, which are used to obtain the index for some chemically significant molecular graphs.

Author Biographies

Sourav Mondal, National Institute of Technology Durgapur.

Dept. of mathematics.

Muhammad Arfan Ali , Virtual University of Pakistan.

Dept. of Mathematics.

Nilanjan De , Calcutta Institute of Engineering and Management.

Dept. of Basic Sciences & Humanities.

Anita Pal, National Institute of Technology Durgapur.

Dept. of mathematics.

References

A. R. Ashrafi, T. Došli?, and A. Hamzeh, “The Zagreb coindices of graph operations”, Discrete applied mathematics, vol. 158, no. 15, pp. 1571–1578, Aug. 2010, doi: 10.1016/j.dam.2010.05.017

M. Azari, “Sharp lower bounds on the Narumi-Katayama index of graph operations”, Applied mathematics and computation, vol. 239, pp. 409-421, Jul. 2014, doi: 10.1016/j.amc.2014.04.088

R. Bhatia and C. Davis, “A Better Bound on the Variance”, The american mathematical monthly, vol. 107, no. 4, pp. 353–357, Apr. 2000, doi: 10.2307/2589180

Z. Che and Z. Chen, “Lower and upper bounds of the forgotten topological index”, MATCH communications in mathematical and in computer chemistry, vol. 76, no. 3, pp. 635-648, 2016. [On line]. Available: https://bit.ly/3hvdXD4

K. C. Das, “On geometrical-arithmetic index of graphs”, MATCH communications in mathematical and in computer chemistry, vol. 64, no. 3, pp. 619-630, 2010. [On line]. Available: https://bit.ly/2N0M9sl

K. C. Das, A. Yurttas, M. Togan, A. Cevik, and I. Cangul, “The multiplicative Zagreb indices of graph operations”, Journal of inequalities and applications, vol. 2013 no. 1, Art ID. 90, Mar. 2013, doi: 10.1186/1029-242x-2013-90

N. De, “Some bounds of reformulated Zagreb indices”, Applied mathematical sciences, vol. 6, no. 101, pp. 5005-5012, 2012. [On line]. Available: https://bit.ly/3db09ds

N. De, S. M. A. Nayeem, and A. Pal, “F-Index of some graph operations”, Discrete mathematics, algorithms and applications, vol. 8, no. 2, Art ID. 1650025, 2006, doi: 10.1142/s1793830916500257

J. B. Diaz and F. T. Metcalf, “Stronger forms of a class of inequalities of G. Pólya-G. Szegö, and L. V. Kantorovich”, Bulletin of the American Mathematical Society, vol. 69, no. 3, pp. 415–419, May 1963, doi: 10.1090/s0002-9904-1963-10953-2

T. Dosli?, “Splices, links and their degree-weighted Wiener polynomials”, Graph Theory Notes New York, vol. 48, pp. 47-55, 2005.

M. Ghorbani and M. A. Hosseinzadeh, “The third version Of Zagreb index”, Discrete mathematics, algorithms and applications, vol. 05, no. 04, Art ID. 1350039, Dec. 2013, doi: 10.1142/S1793830913500390

I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry. Berlin: Springer, 1986, doi: 10.1007/978-3-642-70982-1

I. Gutman and N. Trinajsti?, “Graph theory and molecular orbitals. Total ?-electron energy of alternant hydrocarbons”, Chemical physics letters, vol. 17, no. 4, pp. 535-538, Dec. 1972, doi: 10.1016/0009-2614(72)85099-1

G. H. Hardy, J. E. Littlewood, and P. George, Inequalities, 2nd ed. Cambridge: Cambridge University Press, 1988.

M. H. Khalifeh, H. Yousefi-Azari, and A. R. Ashrafi, “The first and second Zagreb indices of some graph operations”, Discrete applied mathematics, vol. 157, no. 4, pp. 804–811, Feb. 2009, doi:10.1016/j.dam.2008.06.015.

S. Mondal, N. De, and A. Pal, “On neighborhood Zagreb index of product graphs”, May 2018, arXiv:1805.05273

K. Pattabiraman and P. Kandan, “Weighted PI index of corona product of graphs”, Discrete mathematics, algorithms and applications, vol. 06, no. 04, Art ID.1450055, Oct. 2014, doi: 10.1142/S1793830914500554

G. Pólya and G. Szegö, Problems and theorems in analysis. Berlin: Springer, 1972.

N. Trinajsti?, Chemical graph theory. Boca Raton, FL: CRC, 1983.

H. Wiener, “Structural determination of the paraffin boiling points”, Journal of the American Chemical Society, vol. 69, no. 1, pp. 17-20, Jan. 1947, doi: 10.1021/ja01193a005

Published

2020-07-28

How to Cite

[1]
S. Mondal, M. A. Ali, N. De, and A. Pal, “Bounds for neighborhood Zagreb index and its explicit expressions under some graph operations”, Proyecciones (Antofagasta, On line), vol. 39, no. 4, pp. 799-819, Jul. 2020.