آمار
| تعداد نشریات | 6 |
| تعداد شمارهها | 123 |
| تعداد مقالات | 1,565 |
| تعداد مشاهده مقاله | 1,684,055 |
| تعداد دریافت فایل اصل مقاله | 1,570,868 |
The minimum Zagreb indices for unicyclic graphs with fixed Roman domination number | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 14، دوره 11، شماره 3، آذر 2026، صفحه 959-973 اصل مقاله (413.24 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.29846.2188 | ||
| نویسندگان | ||
| Fateme Movahedi1؛ Ayu Ameliatul Shahilah Ahmad Jamri2؛ Roslan Hasni* 2؛ Mohammadhadi Akhbari3؛ Hailiza Kamarulhaili4 | ||
| 1Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan, Iran | ||
| 2Special Interest Group on Modelling and Data Analytics (SIGMDA), Faculty of Computer Science and Mathematics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia | ||
| 3Department of Mathematics, Estahban Branch, Islamic Azad University, Estahban, Iran | ||
| 4School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia | ||
| چکیده | ||
| Let $G=(V, E)$ be a graph with the vertex set $V$ and the edge set $E$. The first Zagreb index of a graph $G$ is defined to be the sum of squares of degrees of all the vertices of the graph. The second Zagreb index of the graph $G$ is the sum of the $d(u)d(v)$ for every edge $uv \in E$, where $d(u)$ and $d(v)$ denote the degree of the vertices $u, v \in V$. In this paper, we propose new lower bounds of the Zagreb indices of unicyclic graphs in terms of the order and the Roman domination number. We prove that $4n-2\left(\gamma_{R}-\left\lceil\dfrac{2n}{3}\right\rceil\right)$ and $4n-3\left(\gamma_{R}-\left\lceil\dfrac{2n}{3}\right\rceil\right)$ are the sharp lower bounds for the first Zagreb index and the second Zagreb index, respectively. Also, we characterize the extremal trees for these lower bounds. | ||
| کلیدواژهها | ||
| Roman domination number؛ Unicyclic graphs؛ Zagreb indices | ||
| مراجع | ||
|
[1] 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
[2] 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
[3] 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
[4] A. Ali, K.C. Das, and S. Akhter, On the extremal graphs for second Zagreb index with fixed number of vertices and cyclomatic number, Miskolc Math. Notes 23 (2022), no. 1, 41–50. http://dx.doi.org/10.18514/MMN.2022.2382
[5] A. Behtoei, M. Jannesari, and B. Taeri, Maximum Zagreb index, minimum hyper-Wiener index and graph connectivity, Appl. Math. Lett. 22 (2009), no. 10, 1571–1576. https://doi.org/10.1016/j.aml.2009.05.001
[6] 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
[7] B. Borovićanin, B. Furtula, and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78 (2017), 17–100.
[8] E.J. Cockayne, P.A. Dreyer Jr, S.M. Hedetniemi, and S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004), no. 1-3, 11–22. https://doi.org/10.1016/j.disc.2003.06.004
[9] K.C. Das, I. Gutman, and B. Zhou, New upper bounds on Zagreb indices, J. Math. Chem. 46 (2009), no. 2, 514–521. https://doi.org/10.1007/s10910-008-9475-3
[10] K.C. Das, K. Xu, and J. Nam, Zagreb indices of graphs, Front. Math. China 10 (2015), no. 3, 567–582. https://doi.org/10.1007/s11464-015-0431-9
[11] 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 Math. 7 (2022), no. 7, 11801–11812.
[12] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total $\varphi$-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538. https://doi.org/10.1016/0009-2614(72)85099-1
[13] M. Liu and B. Liu, The second Zagreb indices and Wiener polarity indices of trees with given degree sequences, MATCH Commun. Math. Comput. Chem. 67 (2012), no. 2, 439.
[14] M. Liu and B. Liu, The second Zagreb indices of unicyclic graphs with given degree sequences, Discrete Appl. Math. 167 (2014), 217–221. https://doi.org/10.1016/j.dam.2013.10.033
[15] 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
[16] A. Poureidi, N.A.A. Aziz, N. Jafari Rad, and H. Kamarulhaili, Computing strong Roman domination of trees and unicyclic graphs in linear time, Bull. Malays. Math. Sci. Soc. 45 (2022), no. 5, 2509–2523. https://doi.org/10.1007/s40840-022-01301-4
[17] S.S. Shirkol, P.P. Kumbargoudra, and M.M. Kaliwal, On Roman domination in middle graphs, J. Shanghai Jiaotong Univ. 17 (2021), no. 7, 10–18.
[18] S.D. Stankov, M.M. Matejić, I.Z. Milovanović, E.I. Milovanović, and S.B. Bozkurt Altindağ, Some new bounds on the first Zagreb index, Electron. J. Math. 1 (2021), 101–107.
[19] D. Thakkar and S. Badiyani, Total Roman domination number of graphs, TWMS J. Apl. Eng. Math. 12 (2022), no. 2, 445–455.
[20] D.B. West, Introduction to Graph Theory, Prentice hall Upper Saddle River, 2001.
[21] K. Xu, The Zagreb indices of graphs with a given clique number, Appl. Math. Lett. 24 (2011), no. 6, 1026– 1030. https://doi.org/10.1016/j.aml.2011.01.034
[22] Z. Yan, H. Liu, and H. Liu, Sharp bounds for the second Zagreb index of unicyclic graphs, J. Math. Chem. 42 (2007), no. 3, 565–574. https://doi.org/10.1007/s10910-006-9132-7
[23] B. Zhou, Zagreb indices, MATCH Commun. Math. Comput. Chem. 52 (2004), no. 1, 113–118. | ||
|
آمار تعداد مشاهده مقاله: 273 تعداد دریافت فایل اصل مقاله: 309 |
||