Minimum atom-bond sum-connectivity index of trees with a fixed order and/or number of pendent vertices

Tariq A. Alraqad; Igor Ž. Milovanović; Hicham Saber; Akbar Ali; Jaya P. Mazorodze; Adel A. Attiya; Tariq A. Alraqad; Igor Ž. Milovanović; Hicham Saber; Akbar Ali; Jaya P. Mazorodze; Adel A. Attiya

doi:10.3934/math.2024182

AIMS Mathematics

2024, Volume 9, Issue 2: 3707-3721. doi: 10.3934/math.2024182

Previous Article Next Article

Research article Special Issues

Minimum atom-bond sum-connectivity index of trees with a fixed order and/or number of pendent vertices

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 55473, Saudi Arabia
2.
Faculty of Electronic Engineering, University of Niš, Niš, Serbia
3.
Department of Mathematics University of Zimbabwe, Harare, Zimbabwe
4.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 19 October 2023 Revised: 15 November 2023 Accepted: 27 November 2023 Published: 09 January 2024
MSC : 05C07, 05C90

Let $ d_u $ be the degree of a vertex $ u $ of a graph $ G $. The atom-bond sum-connectivity (ABS) index of a graph $ G $ is the sum of the numbers $ (1-2(d_v+d_w)^{-1})^{1/2} $ over all edges $ vw $ of $ G $. This paper gives the characterization of the graph possessing the minimum ABS index in the class of all trees of a fixed number of pendent vertices; the star is the unique extremal graph in the mentioned class of graphs. The problem of determining graphs possessing the minimum ABS index in the class of all trees with $ n $ vertices and $ p $ pendent vertices is also addressed; such extremal trees have the maximum degree $ 3 $ when $ n\ge 3p-2\ge7 $, and the balanced double star is the unique such extremal tree for the case $ p = n-2 $.
- topological index,
- atom-bond sum-connectivity,
- tree
Citation: Tariq A. Alraqad, Igor Ž. Milovanović, Hicham Saber, Akbar Ali, Jaya P. Mazorodze, Adel A. Attiya. Minimum atom-bond sum-connectivity index of trees with a fixed order and/or number of pendent vertices[J]. AIMS Mathematics, 2024, 9(2): 3707-3721. doi: 10.3934/math.2024182

Related Papers:

Abstract

Let $ d_u $ be the degree of a vertex $ u $ of a graph $ G $. The atom-bond sum-connectivity (ABS) index of a graph $ G $ is the sum of the numbers $ (1-2(d_v+d_w)^{-1})^{1/2} $ over all edges $ vw $ of $ G $. This paper gives the characterization of the graph possessing the minimum ABS index in the class of all trees of a fixed number of pendent vertices; the star is the unique extremal graph in the mentioned class of graphs. The problem of determining graphs possessing the minimum ABS index in the class of all trees with $ n $ vertices and $ p $ pendent vertices is also addressed; such extremal trees have the maximum degree $ 3 $ when $ n\ge 3p-2\ge7 $, and the balanced double star is the unique such extremal tree for the case $ p = n-2 $.

References

[1]	J. L. Gross, J. Yellen, M. Anderson, Graph theory and its applications, New York: CRC Press, 2018. https://doi.org/10.1201/9780429425134
[2]	S. Wagner, H. Wang, Introduction to chemical graph theory, Boca Raton: CRC Press, 2018.
[3]	M. Randić, On characterization of molecular branching, J. Am. Chem. Soc., 97 (1975), 6609–6615. https://doi.org/10.1021/ja00856a001 doi: 10.1021/ja00856a001
[4]	I. Gutman, Degree-based topological indices, Croat. Chem. Acta, 86 (2013), 351–361. http://dx.doi.org/10.5562/cca2294 doi: 10.5562/cca2294
[5]	X. Li, Y. Shi, A survey on the Randić index, MATCH Commun. Math. Comput. Chem., 59 (2008), 127–156.
[6]	M. Randić, The connectivity index 25 years after, J. Mol. Graph. Model., 20 (2001), 19–35. https://doi.org/10.1016/S1093-3263(01)00098-5 doi: 10.1016/S1093-3263(01)00098-5
[7]	I. Gutman, B. Furtula, Recent results in the theory of Randić index, Math. Chem. Monogr., 2008.
[8]	X. Li, I. Gutman, Mathematical aspects of Randić-type molecular structure descriptors, Univ. Kragujevac, 2006.
[9]	E. Estrada, L. Torres, L. Rodríguez, I. Gutman, An atom-bond connectivity index: Modelling the enthalpy of formation of alkanes, Indian J. Chem. Sec. A, 37 (1998), 849–855.
[10]	E. Estrada, Atom-bond connectivity and the energetic of branched alkanes, Chem. Phys. Lett., 463 (2008), 422–425. https://doi.org/10.1016/j.cplett.2008.08.074 doi: 10.1016/j.cplett.2008.08.074
[11]	B. Zhou, N. Trinajstić, On a novel connectivity index, J. Math. Chem., 46 (2009), 1252–1270. https://doi.org/10.1007/s10910-008-9515-z doi: 10.1007/s10910-008-9515-z
[12]	A. Ali, K. C. Das, D. Dimitrov, B. Furtula, Atom-bond connectivity index of graphs: A review over extremal results and bounds, Discrete Math. Lett., 5 (2021), 68–93. http://dx.doi.org/10.47443/dml.2020.0069 doi: 10.47443/dml.2020.0069
[13]	A. Ali, L. Zhong, I. Gutman, Harmonic index and its generalization: Extremal results and bounds, MATCH Commun. Math. Comput. Chem., 81 (2019), 249–311.
[14]	A. Ali, B. Furtula, I. Redžepović, I. Gutman, Atom-bond sum-connectivity index, J. Math. Chem., 60 (2022), 2081–2093. https://doi.org/10.1007/s10910-022-01403-1 doi: 10.1007/s10910-022-01403-1
[15]	Y. Tang, D. B. West, B. Zhou, Extremal problems for degree-based topological indices, Discrete Appl. Math., 203 (2016), 134–143. https://doi.org/10.1016/j.dam.2015.09.011 doi: 10.1016/j.dam.2015.09.011
[16]	A. Ali, I. Gutman, I. Redžepović, Atom-bond sum-connectivity index of unicyclic graphs and some applications, Electron. J. Math., 5 (2023), 1–7. https://doi.org/10.47443/ejm.2022.039 doi: 10.47443/ejm.2022.039
[17]	J. A. Bondy, U. S. R. Murty, Graph theory, Springer, 2008.
[18]	J. Du, X. Sun, On bond incident degree index of chemical trees with a fixed order and a fixed number of leaves, Appl. Math. Comp., 464 (2024), 128390. https://doi.org/10.1016/j.amc.2023.128390 doi: 10.1016/j.amc.2023.128390
[19]	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. https://doi.org/10.47443/dml.2023.016

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)