آمار
| تعداد نشریات | 7 |
| تعداد شماره‌ها | 102 |
| تعداد مقالات | 1,142 |
| تعداد مشاهده مقاله | 1,050,530 |
| تعداد دریافت فایل اصل مقاله | 870,861 |
A note on the re-defined third Zagreb index of trees | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 08 دی 1402 اصل مقاله (319.58 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.28868.1757 | ||
| نویسنده | ||
| Nasrin Dehgardi* | ||
| Sirjan University of Technology, Sirjan 78137, Iran | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Zagreb indices؛ re-defined third Zagreb index؛ trees | ||
| مراجع | ||
|
[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.
[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.
| ||
|
آمار تعداد مشاهده مقاله: 142 تعداد دریافت فایل اصل مقاله: 346 |
||