A ‎note‎ ‎on‎ ‎the‎ ‎r‎e-defined third Zagreb index of trees

Document Type : Original paper

Author

Sirjan University of Technology, Sirjan 78137, Iran

Abstract

For a graph $\Gamma$‎, ‎the re-defined third Zagreb index is defined as $$ReZG_3(\Gamma)=\sum_{ab\in E(\Gamma)}\deg_\Gamma(a) ‎\deg_\Gamma(b)\Big(‎\deg_\Gamma(a)+‎\deg_\Gamma(b)\Big)‎‎,$$‎
‎where $\deg_\Gamma(a)$ is the degree of‎ ‎vertex $a$‎. ‎We prove for any tree $T$ with $n$ vertices and maximum degree $\Delta$‎, ‎‎$ReZG_3(T)\geq‎16n+\Delta^3+2\Delta^2-13\Delta-26$ ‎when ‎‎$‎\Delta< n-1‎$ ‎and‎ 
$ReZG_3(T)=‎n\Delta^2+n\Delta-\Delta^2-\Delta$ ‎when ‎‎$‎\Delta=n-1‎$. 
‎Also we determine the corresponding extremal trees‎. ‎‎

Keywords

Main Subjects

References

[1] 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. Optim. 8 (2023), no. 2, 391–396.
https://doi.org/10.22049/cco.2022.27553.1288
[2] A. Ali, I. Gutman, E. Milovanovi´c, and I. Milovanovi´c, Sum of powers of the degrees of graphs: Extremal results and bounds, MATCH Commun. Math. Comput. Chem. 80 (2018), no. 1, 5–84.
[3] A. Alwardi, A. Alqesmah, R. Rangarajan, and I.N. Cangul, Entire Zagreb indices of graphs, Discrete Math. Algorithms Appl. 10 (2018), no. 3, Article ID:1850037.
https://doi.org/10.1142/S1793830918500374
[4] M. Azari, N. Dehgardi, and T. Došlić, Lower bounds on the irregularity of trees and unicyclic graphs, Discrete Appl. Math. 324 (2023), 136–144.
https://doi.org/10.1016/j.dam.2022.09.022
[5] B. Borovićanin, B. Furtula, and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78 (2017), 17–100.
[6] N. Dehgardi and H. Aram, Sharp bounds on the augmented Zagreb index of graph operations, Kragujevac J. Math. 44 (2020), no. 6, 509–522.
http://doi.org/10.46793/KgJMat2004.509D
[7] N. Dehgardi and T. Došlić, Lower bounds on the general first Zagreb index of graphs with low cyclomatic number, Discrete Appl. Math. 345 (2024), 52–61.
https://doi.org/10.1016/j.dam.2023.11.033
[8] N. Dehgardi and J.B. Liu, Lanzhou index of trees with fixed maximum degree, MATCH Commun. Math. Comput. Chem. 86 (2021), 3–10.
[9] I. Gutman, Multiplicative Zagreb indices of trees, Bull. Int. Math. Virt. Instit. 18 (2011), 17–23.
[10] I. Gutman, B. Furtula, Ž.K. Vukićević, and G. Popivoda, On Zagreb indices and coindices, MATCH Commun. Math. Comput. Chem. 74 (2015), 5–16.
[11] I. Gutman, E. Milovanović, and I. Milovanović, Beyond the Zagreb indices, AKCE Int. J. Graphs Comb. 17 (2020), no. 1, 74–85.
https://doi.org/10.1016/j.akcej.2018.05.002
[12] I. Gutman, B. Rušˇcić, N. Trinajstić, and C.F. Wilcox Jr, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys. 62 (1975), no. 9, 3399–3405.
https://doi.org/10.1063/1.430994
[13] I. Gutman, Z. Shao, Z. Li, S.S. Wang, and P. We, Leap Zagreb indices of trees and unicyclic graphs, Commun. Comb. Optim. 3 (2018), no. 2, 179–194.
https://doi.org/10.22049/cco.2018.26285.1092
[14] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538.
https://doi.org/10.1016/0009-2614(72)85099-1
[15] L. Luo, N. Dehgardi, and A. Fahad, Lower bounds on the entire Zagreb indices of trees, Discrete Dyn. Nat. Soc. 2020 (2020), Article ID: 8616725.
https://doi.org/10.1155/2020/8616725
[16] Y. Ma, S. Cao, Y. Shi, M. Dehmer, and C. Xia, Nordhaus-Gaddum type results for graph irregularities, Appl. Math. Compute. 343 (2019), 268–272.
https://doi.org/10.1016/j.amc.2018.09.057
[17] A. Usha P.S. Ranjini, V. Lokesha, Relation between phynylene and hexagonal squeez using harmonic index, Int. J. Graph Theory 1 (2013), 116–121.
[18] P.S. Ranjini, A. Usha, V. Lokesha, and T. Deepika, Hormonic index, redefined Zagreb indices of dragon graph with complete graph, Asian J. Math. Comput. Res. 9 (2016), 161–166.
[19] R. Rasi, S.M. Sheikholeslami, and A. Behmaram, An upper bound on the first Zagreb index and coindex in trees, Iranian J. Math. Chem. 8 (2017), no. 1, 71–82.
https://doi.org/10.22052/ijmc.2017.42995
[20] D. Vukicevic, Q. Li, J. Sedlar, and T. Došlić, Lanzhou index, MATCH Commun. Math. Comput. Chem. 80 (2018), no. 3, 863–876.
[21] K. Xu and H. Hua, A unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs, MATCH Commun. Math. Comput. Chem. 68 (2012), 241–256.
 
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 29 December 2023

Files

Share

How to cite

Statistics