Let $ G = (V(G), E(G)) $ be a simple connected graph with vertex set $ V(G) $ and edge set $ E(G) $. The atom-bond sum-connectivity (ABS) index was proposed recently and is defined as $ ABS(G) = \sum_{uv\in E(G)}\sqrt{\frac{d_{G}(u)+d_{G}(v)-2}{d_{G}(u)+d_{G}(v)}} $, where $ d_{G}(u) $ represents the degree of vertex $ u\in V(G) $. A connected graph $ G $ is called a unicyclic graph if $ |V(G)| = |E(G)| $. In this paper, we determine the maximum ABS index of unicyclic graphs with given diameter. In addition, the corresponding extremal graphs are characterized.
Citation: Zhen Wang, Kai Zhou. On the maximum atom-bond sum-connectivity index of unicyclic graphs with given diameter[J]. AIMS Mathematics, 2024, 9(8): 22239-22250. doi: 10.3934/math.20241082
Let $ G = (V(G), E(G)) $ be a simple connected graph with vertex set $ V(G) $ and edge set $ E(G) $. The atom-bond sum-connectivity (ABS) index was proposed recently and is defined as $ ABS(G) = \sum_{uv\in E(G)}\sqrt{\frac{d_{G}(u)+d_{G}(v)-2}{d_{G}(u)+d_{G}(v)}} $, where $ d_{G}(u) $ represents the degree of vertex $ u\in V(G) $. A connected graph $ G $ is called a unicyclic graph if $ |V(G)| = |E(G)| $. In this paper, we determine the maximum ABS index of unicyclic graphs with given diameter. In addition, the corresponding extremal graphs are characterized.
| [1] |
K. Aarthi, S. Elumalai, S. Balachandran, S. Mondal, Extremal values of the atom-bond sum-connectivity index in bicyclic graphs, J. Appl. Math. Comput., 69 (2023), 4269–4285. http://dx.doi.org/10.1007/s12190-023-01924-1 doi: 10.1007/s12190-023-01924-1
|
| [2] |
A. Ali, B. Furtula, I. Redžepović, I. Gutman, Atom-bond sum-connectivity index, J. Math. Chem., 60 (2022), 2081–2093. http://dx.doi.org/10.1007/s10910-022-01403-1 doi: 10.1007/s10910-022-01403-1
|
| [3] |
A. Ali, I. Gutman, I. Redžepović, Atom-bond sum-connectivity index of unicyclic graphs and some applications, Electron. J. Math., 5 (2023), 1–7. http://dx.doi.org/10.47443/ejm.2022.039 doi: 10.47443/ejm.2022.039
|
| [4] |
A. Ali, I. Gutman, I. Redžepović, J. Mazorodze, A. Albalahi, A. Hamza, On the difference of atom-bond sum-connectivity and atom-bond-connectivity indices, MATCH Commun. Math. Comput. Chem., 91 (2024), 725–740. http://dx.doi.org/10.46793/match.91-3.725A doi: 10.46793/match.91-3.725A
|
| [5] |
T. Alraqad, I. Milovanović, H. Saber, A. Ali, J. Mazorodze, A. Attiya, Minimum atom-bond sum-connectivity index of trees with a fixed order and/or number of pendent vertices, AIMS Mathematics, 9 (2024), 3707–3721. http://dx.doi.org/10.3934/math.2024182 doi: 10.3934/math.2024182
|
| [6] |
Z. Du, B. Zhou, N. Trinajstić, Minimum sum-connectivity indices of trees and unicyclic graphs of a given matching number, J. Math. Chem., 47 (2010), 842–855. http://dx.doi.org/10.1007/s10910-009-9604-7 doi: 10.1007/s10910-009-9604-7
|
| [7] | E. Estrada, L. Torres, L. Rodríguez, I. Gutman, An atom-bond connectivity index: modelling the enthalpy of formation of alkanes, Indian J. Chem., 37A (1998), 849–855. |
| [8] |
E. Estrada, Atom-bond connectivity and the energetic of branched alkanes, Chem. Phys. Lett., 463 (2008), 422–425. http://dx.doi.org/10.1016/j.cplett.2008.08.074 doi: 10.1016/j.cplett.2008.08.074
|
| [9] | Y. Gao, Y. Shao, The smallest ABC index of trees with $n$ pendent vertices, MATCH Commun. Math. Comput. Chem., 76 (2016), 141–158. |
| [10] | I. Gutman, J. Tosovic, S. Radenkovic, S. Markovic, On atom-bond connectivity index and its chemical applicability, Indian J. Chem., 51A (2012), 690–694. |
| [11] |
Y. Hu, F. Wang, On the maximum atom-bond sum-connectivity index of trees, MATCH Commun. Math. Comput. Chem., 91 (2024), 709–723. http://dx.doi.org/10.46793/match.91-3.709H doi: 10.46793/match.91-3.709H
|
| [12] |
P. Nithya, S. Elumalai, S. Balachandran, S. Mondal, Smallest ABS index of unicyclic graphs with given girth, J. Appl. Math. Comput., 69 (2023), 3675–3692. http://dx.doi.org/10.1007/s12190-023-01898-0 doi: 10.1007/s12190-023-01898-0
|
| [13] |
F. Li, Q. Ye, H. Lu, The greatest values for atom-bond sum-connectivity index of graphs with given parameters, Discrete Appl. Math., 344 (2024), 188–196. http://dx.doi.org/10.1016/j.dam.2023.11.029 doi: 10.1016/j.dam.2023.11.029
|
| [14] |
S. Noureen, A. Ali, Maximum atom-bond sum-connectivity index of $n$-order trees with fixed number of leaves, Discrete Math. Lett., 12 (2023), 26–28. http://dx.doi.org/10.47443/dml.2023.016 doi: 10.47443/dml.2023.016
|
| [15] |
Z. Shao, P. Wu, Y. Gao, I. Gutman, X. Zhang, On the maximum ABC index of graphs without pendent vertices, Appl. Math. Comput., 315 (2017), 298–312. http://dx.doi.org/10.1016/j.amc.2017.07.075 doi: 10.1016/j.amc.2017.07.075
|
| [16] |
R. Xing, B. Zhou, N. Trinajstić, Sum-connectivity index of molecular trees, J. Math. Chem., 48 (2010), 583–591. http://dx.doi.org/10.1007/s10910-010-9693-3 doi: 10.1007/s10910-010-9693-3
|
| [17] |
Y. Zhang, H. Wang, G. Su, K. Das, Extremal problems on the Atom-bond sum-connectivity indices of trees with given matching number or domination number, Discrete Appl. Math., 345 (2024), 190–206. http://dx.doi.org/10.1016/j.dam.2023.11.046 doi: 10.1016/j.dam.2023.11.046
|
| [18] |
B. Zhou, N. Trinajstić, On a novel connectivity index, J. Math. Chem., 46 (2009), 1252–1270. http://dx.doi.org/10.1007/s10910-008-9515-z doi: 10.1007/s10910-008-9515-z
|
| [19] |
X. Zuo, A. Jahanbani, H. Shooshtari, On the atom-bond sum-connectivity index of chemical graphs, J. Mol. Struct., 1296 (2024), 136849. http://dx.doi.org/10.1016/j.molstruc.2023.136849 doi: 10.1016/j.molstruc.2023.136849
|
Figures(2)
Zhen Wang, Kai Zhou. On the maximum atom-bond sum-connectivity index of unicyclic graphs with given diameter[J]. AIMS Mathematics, 2024, 9(8): 22239-22250. doi: 10.3934/math.20241082
DownLoad: