Opuscula Math. 39, no. 3 (2019), 433-446
https://doi.org/10.7494/OpMath.2019.39.3.433

Opuscula Mathematica

# General multiplicative Zagreb indices of graphs with given clique number

Tomáš Vetrík
Selvaraj Balachandran

Abstract. We obtain lower and upper bounds on general multiplicative Zagreb indices for graphs of given clique number and order. Bounds on the basic multiplicative Zagreb indices and on the multiplicative sum Zagreb index follow from our results. We also determine graphs with the smallest and the largest indices.

Keywords: clique number, multiplicative Zagreb index, chromatic number.

Mathematics Subject Classification: 05C35, 05C07.

Full text (pdf)

1. B. Basavanagoud, S. Patil, Multiplicative Zagreb indices and coindices of some derived graphs, Opuscula Math. 36 (2016) 3, 287-299.
2. P. Erdős, On the graph theorem of Turán, Mat. Lapok 21 (1970), 249-251.
3. M. Ghorbani, M. Songhori, Computing multiplicative Zagreb indices with respect to chromatic and clique numbers, Iran. J. Math. Chem. 5 (2014) 1, 11-18.
4. R. Kazemi, Note on the multiplicative Zagreb indices, Discrete Appl. Math. 198 (2016), 147-154.
5. J. Liu, Q. Zhang, Sharp upper bounds for multiplicative Zagreb indices, MATCH Commun. Math. Comput. Chem. 68 (2012) 1, 231-240.
6. E.F. Nezhad, A. Iranmanesh, A. Tehranian, M. Azari, Strict lower bounds on the multiplicative Zagreb indices of graph operations, Ars Combin. 117 (2014), 399-409.
7. T. Vetrík, S. Balachandran, General multiplicative Zagreb indices of trees, Discrete Appl. Math. 247 (2018), 341-351.
8. C. Wang, J.-B. Liu, S. Wang, Sharp upper bounds for multiplicative Zagreb indices of bipartite graphs with given diameter, Discrete Appl. Math. 227 (2017), 156-165.
9. K. Xu, K.C. Das, On Harary index of graphs, Discrete Appl. Math. 159 (2011) 15, 1631-1640.
10. K. Xu, K. Tang, K.C. Das, H. Yue, Chromatic number and some multiplicative vertex-degree-based indices of graphs, Kragujevac J. Math. 36 (2012) 2, 323-333.
• Selvaraj Balachandran
• https://orcid.org/0000-0003-2782-6315
• University of the Free State, Department of Mathematics and Applied Mathematics, P.O. Box 339, 9300 Bloemfontein, South Africa
• SASTRA Deemed University, Department of Mathematics, School of Humanities and Sciences, 613402 Thanjavur, Tamil Nadu, India
• Communicated by Mirko Horňák.
• Accepted: 2018-11-21.
• Published online: 2019-02-23.