Extremal Zagreb indices of a tree with given double Roman domination number

Articles in Press

Document Type : Original paper

Authors

Department of Mathematics, Tafresh University, Tafresh 39518-79611, Iran

Abstract

In this article, some upper and lower bounds for the first and the second Zagreb indices of an arbitrary tree in terms of its order and double Roman domination number $\gamma_{dR}$, (depending on whether $\gamma_{dR}$ is odd or even), are stated. Also, all extremal trees attaining equality are characterized.

Keywords

Main Subjects

References

[1] H.A. Ahangar, M. Chellali, and S.M. Sheikholeslami, On the double Roman domination in graphs, Discrete Appl. Math. 232 (2017), 1–7. https://doi.org/10.1016/j.dam.2017.06.014
[2] A.A.S. Ahmad Jamri, R. Hasni, M. Kamran Jamil, and D.A. Mojdeh, Maximum second Zagreb index of trees with given Roman domination number, Trans. Comb. 12 (2023), no. 1, 1–10. https://doi.org/10.22108/toc.2022.128323.1848
[3] A.A.S. Ahmad Jamri, R. Hasni, and S.K. Said Husain, On the Zagreb indices of graphs with given Roman domination number, Commun. Comb. Optim. 8 (2023), no. 1, 141–152. https://doi.org/10.22049/cco.2021.27439.1263
[4] A.A.S. Ahmad Jamri, F. Movahedi, R. Hasni, and M.H. Akhbari, A lower bound for the second Zagreb index of trees with given Roman domination number, Commun. Comb. Cptim. 8 (2023), no. 2, 391–396. https://doi.org/10.22049/cco.2022.27553.1288
[5] R.A. Beeler, T.W. Haynes, and S.T. Hedetniemi, Double Roman domination, Discrete Appl. Math. 211 (2016), 23–29.
https://doi.org/10.1016/j.dam.2016.03.017
[6] B. Borovićanin, K.C. Das, B. Furtula, and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78 (2017), no. 1, 17–100.
[7] B. Borovićanin and B. Furtula, On extremal Zagreb indices of trees with given domination number, Appl. Math. Comput. 279 (2016), 208–218. https://doi.org/10.1016/j.amc.2016.01.017
[8] Z. Du, A.A.S. Ahmad Jamri, R. Hasni, and D.A. Mojdeh, Maximal first Zagreb index of trees with given Roman domination number, AIMS Maths. 7 (2021), no. 7, 11801–11812. https://doi.org/10.3934/math.2022658
[9] I. Gutman and K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), no. 1, 83–92.
[10] I. Gutman and N. Trinajsti´c, Graph theory and molecular orbitals. Total $\pi$-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538. https://doi.org/10.1016/0009-2614(72)85099-1
[11] N. Jafari Rad and H. Rahbani, Some progress on the double Roman domination in graphs, Discuss. Math. Graph Theory 39 (2018), no. 1, 41–53. http://doi.org/10.7151/dmgt.2069
[12] D.A. Mojdeh, M. Habibi, L. Badakhshian, and Y. Rao, Zagreb indices of trees, unicyclic and bicyclic graphs with given (total) domination, IEEE Access 7 (2019), 94143–94149. https://doi.org/10.1109/ACCESS.2019.2927288
[13] S. Nikolić, G. Kovačević, A. Miličević, and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta. 76 (2003), no. 2, 113–124.
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 03 September 2025

Files

Share

How to cite

Statistics