Nano-Zagreb Index and Multiplicative Nano-Zagreb Index of Some Graph Operations

Akbar Jahanbani, Hajar Shooshtary


Let G be a graph with vertex set V(G) and edge set E(G). The Nano-Zagreb and multiplicative Nano-Zagreb indices of G are NZ(G) = \prod_{uv \in E(G)} (d^2(u) - d^2(v)) and N*Z(G) = \prod_{uv \in E(G)} (d^2(u) - d^2(v)), respectively, where d(v) is the degree of the vertex v. In this paper, we define two types of Zagreb indices based on degrees of vertices. Also the Nano-Zagreb index and multiplicative Nano-Zagreb index of the Cartesian product, symmetric difference, composition and disjunction of graphs are computed.


Graph operations‎; ‎Nano-Zagreb index‎; ‎Multiplicative‎ ‎Nano-Zagreb index‎; ‎Zagreb index‎

Full Text:



I. Gutman and N. Trinajstic, “Graph theory and molecular orbitals. total j-electron energy of alternant hydrocarbons,” Chemical Physics Letters, vol. 17, no. 4, pp. 535–538, 1972.

R. Todeschini and V. Consonni, “New local vertex invariants and molecular descriptors based on functions of the vertex degrees,” MATCH Commun. Math. Comput. Chem, vol. 64, no. 2, pp. 359–372, 2010.

R. Todeschini, D. Ballabio, and V. Consonni, “Novel molecular descriptors based on functions of new vertex degrees,” in Novel Molecular Structure Descriptors - Theory and Applications I. Kragujevac: University of Kragujevac, 2010, pp. 73–100.

M. Eliasi, A. Iranmanesh, and I. Gutman, “Multiplicative versions of first zagreb index,” Match-Communications in Mathematical and Computer Chemistry, vol. 68, no. 1, p. 217, 2012.

I. Gutman, “Multiplicative zagreb indices of trees,” Bull. Soc. Math. Banja Luka, vol. 18, pp. 17–23, 2011.

K. Xu and K. C. Das, “Trees, unicyclic, and bicyclic graphs extremal with respect to multiplicative sum zagreb index,” Match Communications in Mathematical and Computer Chemistry, vol. 68, no. 1, p. 257, 2012.

I. Gutman, “Degree-based topological indices,” Croatica Chemica Acta, vol. 86, no. 4, pp. 351–361, 2013.

I. Gutman, “A formula for the wiener number of trees and its extension to graphs containing cycles,” Graph Theory Notes NY, vol. 27, no. 9, pp. 9–15, 1994.

I. Gutman, S. Klavzar, and B. Mohar, Fifty years of the Wiener index. University, Department of Mathematics, 1997.

H. Wiener, “Structural determination of paraffin boiling points,” Journal of the American Chemical Society, vol. 69, no. 1, pp. 17–20, 1947.

W. Gao, W. Wang, and M. Farahani, “Topological indices study of molecular structure in anticancer drugs,” Journal of Chemistry, vol. 2016, 2016.

Y. Huang, B. Liu, and M. Zhang, “On comparing the variable zagreb indices,” MATCH Commun. Math. Comput. Chem, vol. 63, pp. 453–460, 2010.

B. Liu and Z. You, “A survey on comparing zagreb indices,” MATCH Commun. Math. Comput. Chem, vol. 65, no. 3, pp. 581–593, 2011.

N. Rad, A. Jahanbani, and I. Gutman, “Zagreb energy and zagreb estrada index of graphs,” Match. Commun. Math. Comput. Chem, vol. 79, pp. 371–386, 2018.

I. Gutman and K. C. Das, “The first zagreb index 30 years after,” MATCH Commun. Math. Comput. Chem, vol. 50, no. 1, pp. 83–92, 2004.

W. Imrich and S. Klavzar, Product graphs: structure and recognition. Wiley, 2000.

M. Khalifeh, H. Yousefi-Azari, and A. Ashrafi, “The hyper-Wiener index of graph operations,” Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1402–1407, 2008.

I. Cangul, A. Yurttas, M. Togan, and A. Cevik, “Some formulae for the zagreb indices of graphs,” AIP Conference Proceedings, vol. 1479, no. 1, pp. 365–367, 2012.

K. 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, p. 90, 2013.

V. Kulli, “First multiplicative k banhatti index and coindex of graphs,” Annals of Pure and Applied Mathematics, vol. 11, no. 2, pp. 79–82, 2016.

V. Kulli, “Second multiplicative k banhatti index and coindex of graphs,” Journal of Computer and Mathematical Sciences, vol. 7, no. 5, pp. 254–258, 2016.

J. Steele, “An introduction to the art of mathematical inequalities,” 2004.



  • There are currently no refbacks.

View My Stats

Creative Commons License
International Journal of Computing Science and Applied Mathematics by Department Mathematics ITS is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at