Sharp bounds on additively weighted Mostar index of Cacti

Document Type : Original paper

Authors

1 Department of Mathematics, Bishop Chulaparambil Memorial College(B.C.M), Kottayam

2 Department of Mathematics, St.Aloysius CollegeEdathua, Alappuzha - 689573, India

Abstract

Let C(n, t) denotes the collection of all cacti of order n with exactly t cycles and Ctn

denotes the cacti of order n and t end vertices. In this paper, we compute the upper bound, second largest upper bound, and third largest upper bound of the additively weighted Mostar index of graphs in C(n, t). We also determine the upper bound of the additively weighted Mostar index for cacti of order n with a fixed number of end vertices. We characterize all the graphs attaining the bounds.

Keywords

Main Subjects

References

[1] A. Ali and T. Došlić, Mostar index: Results and perspectives, Appl. Math. Comput. 404 (2021), Article ID: 126245.
https://doi.org/10.1016/j.amc.2021.126245
[2] S. Brezovnik and N. Tratnik, General cut method for computing Szeged-like topological indices with applications to molecular graphs, Int. J. Quantum Chem. 121 (2021), no. 6, Article ID: e26530.
https://doi.org/10.1002/qua.26530
[3] S. Chen, Cacti with the smallest, second smallest, and third smallest Gutman index, J. Comb. Optim. 31 (2016), no. 1, 327–332.
https://doi.org/10.1007/s10878-014-9743-z
[4] K. Deng and S. Li, On the extremal values for the Mostar index of trees with given degree sequence, Appl. Math. Comput. 390 (2021), Article ID: 125598.
https://doi.org/10.1016/j.amc.2020.125598
[5] T. Došlić, I. Martinjak, R. Škrekovski, S.T. Spužević, and I. Zubac, Mostar index, J. Math. Chem. 56 (2018), no. 10, 2995–3013.
https://doi.org/10.1007/s10910-018-0928-z
[6] F. Gao, K. Xu, and T. Došlić, On the difference of Mostar index and irregularity of graphs, Bull. Malays. Math. Sci. Soc. 44 (2021), no. 2, 905–926.
https://doi.org/10.1007/s40840-020-00991-y
[7] F. Hayat and B. Zhou, On cacti with large Mostar index, Filomat 33 (2019), no. 15, 4865–4873.
https://doi.org/10.2298/FIL1915865H
[8] S. Li, H. Yang, and Q. Zhao, Sharp bounds on Zagreb indices of cacti with $k$ pendant vertices, Filomat 26 (2012), no. 6, 1189–1200.
https://doi.org/10.2298/FIL1206189L
[9] A. Lin, R. Luo, and X. Zha, A sharp lower bound of the Randi´c index of cacti with r pendants, Discrete Appl. Math. 156 (2008), no. 10, 1725–1735.
https://doi.org/10.1016/j.dam.2007.08.031
[10] H. Liu, Extremal cacti with respect to Sombor index, Iranian J. Math. Chem. 12 (2021), no. 4, 197–208.
https://doi.org/10.22052/ijmc.2021.243026.1582
[11] H. Liu and M. Lu, A unified approach to extremal cacti for different indices, MATCH Commun. Math. Comput. Chem 58 (2007), no. 1, 183–194.
[12] T. Réti, A. Ali, and I. Gutman, On bond-additive and atoms-pair-additive indices of graphs, Electron. J. Math. 2 (2021), 52–61.
https://doi.org/10.47443/ejm.2021.0033
[13] N. Tratnik, Computing the Mostar index in networks with applications to molecular graphs, Iranian J. Math. Chem. 12 (2021), no. 1, 1–18.
https://doi.org/10.22052/ijmc.2020.240316.1526
[14] C. Wang, S. Wang, and B. Wei, Cacti with extremal PI index, Trans. Comb. 5 (2016), no. 4, 1–8.
https://doi.org/10.22108/toc.2016.14786
[15] D.F. Wang and S.W. Tan, The maximum hyper-Wiener index of cacti, J. Appl. Math. Comput. 47 (2015), no. 1, 91–102.
https://doi.org/10.1007/s12190-014-0763-8
[16] H. Wang and L. Kang, More on the Harary index of cacti, J. Appl. Math. Comput. 43 (2013), no. 1, 369–386.
https://doi.org/10.1007/s12190-013-0668-y
[17] S. Wang, On extremal cacti with respect to the revised Szeged index, Discrete Appl. Math. 233 (2017), 231–239.
https://doi.org/10.1016/j.dam.2017.07.027
[18] S. Wang, On extremal cacti with respect to the Szeged index, Appl. Math. Comput. 309 (2017), 85–92.
https://doi.org/10.1016/j.amc.2017.03.036
[19] Q. Xiao, M. Zeng, Z. Tang, H. Deng, and H. Hua, Hexagonal chains with the first three minimal Mostar indices, MATCH Commun. Math. Comput. Chem 85 (2021), no. 1, 47–61.
[20] F. Yasmeen, S. Akhter, K. Ali, and S.T.R. Rizvi, Edge Mostar indices of cacti graph with fixed cycles, Front. Chem. 9 (2021), Article ID: 693885.
https://doi.org/10.3389/fchem.2021.693885
 
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 05 May 2024

Files

Share

How to cite

Statistics