On the variable inverse sum deg index

Edil D. Molina; Paul Bosch; José M. Sigarreta; Eva Tourís; Edil D. Molina; Paul Bosch; José M. Sigarreta; Eva Tourís

doi:10.3934/mbe.2023387

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 5: 8800-8813. doi: 10.3934/mbe.2023387

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On the variable inverse sum deg index

1.
Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No.54 Col. Garita, 39650 Acalpulco Gro., Mexico
2.
Facultad de Ingeniería, Universidad del Desarrollo, Ave. La Plaza 680, San Carlos de Apoquindo, Las Condes, Santiago, Chile
3.
Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, Madrid 28049, Spain

Academic Editor: Hao Wang

Received: 13 December 2022 Revised: 10 February 2023 Accepted: 15 February 2023 Published: 09 March 2023

Several important topological indices studied in mathematical chemistry are expressed in the following way $ \sum_{uv \in E(G)} F(d_u, d_v) $, where $ F $ is a two variable function that satisfies the condition $ F(x, y) = F(y, x) $, $ uv $ denotes an edge of the graph $ G $ and $ d_u $ is the degree of the vertex $ u $. Among them, the variable inverse sum deg index $ IS\!D_a $, with $ F(d_u, d_v) = 1/(d_u^a+d_v^a) $, was found to have several applications. In this paper, we solve some problems posed by Vukičević ^[1], and we characterize graphs with maximum and minimum values of the $ IS\!D_a $ index, for $ a < 0 $, in the following sets of graphs with $ n $ vertices: graphs with fixed minimum degree, connected graphs with fixed minimum degree, graphs with fixed maximum degree, and connected graphs with fixed maximum degree. Also, we performed a QSPR analysis to test the predictive power of this index for some physicochemical properties of polyaromatic hydrocarbons.
- variable inverse sum deg index,
- inverse sum indeg index,
- optimization on graphs,
- degree-based topological index
Citation: Edil D. Molina, Paul Bosch, José M. Sigarreta, Eva Tourís. On the variable inverse sum deg index[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 8800-8813. doi: 10.3934/mbe.2023387

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Abstract

Several important topological indices studied in mathematical chemistry are expressed in the following way $ \sum_{uv \in E(G)} F(d_u, d_v) $, where $ F $ is a two variable function that satisfies the condition $ F(x, y) = F(y, x) $, $ uv $ denotes an edge of the graph $ G $ and $ d_u $ is the degree of the vertex $ u $. Among them, the variable inverse sum deg index $ IS\!D_a $, with $ F(d_u, d_v) = 1/(d_u^a+d_v^a) $, was found to have several applications. In this paper, we solve some problems posed by Vukičević ^[1], and we characterize graphs with maximum and minimum values of the $ IS\!D_a $ index, for $ a < 0 $, in the following sets of graphs with $ n $ vertices: graphs with fixed minimum degree, connected graphs with fixed minimum degree, graphs with fixed maximum degree, and connected graphs with fixed maximum degree. Also, we performed a QSPR analysis to test the predictive power of this index for some physicochemical properties of polyaromatic hydrocarbons.

References

[1]	D. Vukičević, Bond additive modeling 5. Mathematical properties of the variable sum exdeg index, Croat. Chem. Acta, 84 (2011), 93–101.
[2]	E. Estrada, Quantifying network heterogeneity, Phys. Rev. E, 82 (2010), 066102. https://doi.org/10.1103/PhysRevE.82.066102 doi: 10.1103/PhysRevE.82.066102
[3]	I. Gutman, B. Furtula, V. Katanić, Randić index and information, AKCE Int. J. Graphs Comb., 15 (2018), 307–312. https://doi.org/10.1016/j.akcej.2017.09.006 doi: 10.1016/j.akcej.2017.09.006
[4]	V. R. Kulli, F-Revan index and F-Revan polynomial of some families of benzenoid systems, J. Global Res. Math. Arch., 5 (2018), 1–6.
[5]	V. R. Kulli, Revan indices of oxide and honeycomb networks, Int. J. Math. Appl., 55 (2017), 7.
[6]	A. Miličević, S. Nikolić, On variable Zagreb indices, Croat. Chem. Acta, 77 (2004), 97–101.
[7]	E. D. Molina, J. M. Rodríguez, J. L. Sánchez, J. M. Sigarreta, Some properties of the arithmetic–geometric index, Symmetry, 13 (2021), 857. https://doi.org/10.3390/sym13050857 doi: 10.3390/sym13050857
[8]	J. Pineda, C. Martínez, J. A. Méndez, J. Muños, J. M. Sigarreta, Application of bipartite networks to the study of water quality, Sustainability, 12 (2020), 5143. https://doi.org/10.3390/su12125143 doi: 10.3390/su12125143
[9]	N. Zahra, M. Ibrahim, M. K. Siddiqui, On topological indices for swapped networks modeled by optical transpose interconnection system, IEEE Access, 8 (2020), 200091–200099. https://doi.org10.1109/ACCESS.2020.3034439 doi: 10.1109/ACCESS.2020.3034439
[10]	A. Ali, L. Zhong, I. Gutman, Harmonic index and its generalizations: extremal results and bounds, MATCH Commun. Math. Comput. Chem., 81 (2020), 249–311.
[11]	K. C. Das, On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem., 63 (2010), 433–440.
[12]	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. https://doi.org/10.1007/s10910-009-9604-7 doi: 10.1007/s10910-009-9604-7
[13]	R. Cruz, J. Monsalve, J. Rada, On chemical trees that maximize atombond connectivity index, its exponential version, and minimize exponential geometric-arithmetic index, MATCH Commun. Math. Comput. Chem., 84 (2020), 691–718.
[14]	R. Cruz, J. Monsalve, J. Rada, Trees with maximum exponential Randic index, Discrete Appl. Math., 283 (2020), 634–643. https://doi.org/10.1016/j.dam.2020.03.009 doi: 10.1016/j.dam.2020.03.009
[15]	R. Cruz, J. Rada, Extremal values of exponential vertex-degree-based topological indices over graphs, Kragujevac J. Math., 46 (2022), 105–113.
[16]	K. C. Das, Y. Shang, Some extremal graphs with respect to sombor index, Mathematics, 9 (2021), 1202. https://doi.org/10.3390/math9111202 doi: 10.3390/math9111202
[17]	M. A. Iranmanesh, M. Saheli, On the harmonic index and harmonic polynomial of Caterpillars with diameter four, Iran. J. Math. Chem., 5 (2014), 35–43. https://doi.org/10.22052/IJMC.2015.9044 doi: 10.22052/IJMC.2015.9044
[18]	X. Li, I. Gutman, Mathematical aspects of Randić-type molecular structure descriptors, Croat. Chem. Acta, 79 (2006).
[19]	D. Vukičević, M. Gašperov, Bond additive modeling 1. Adriatic indices, Croat. Chem. Acta, 83 (2010), 243–260.
[20]	D. Vukičević, Bond additive modeling 2. Mathematical properties of max-min rodeg index, Croat. Chem. Acta, 83 (2010), 261–273.
[21]	W. Carballosa, J. A. Méndez-Bermúdez, J. M. Rodríguez, J. M. Sigarreta, Inequalities for the variable inverse sum deg index, Submitted.
[22]	H. Chen, H. Deng, The inverse sum indeg index of graphs with some given parameters, Discr. Math. Algor. Appl., 10 (2018), 1850006. https://doi.org/10.1142/S1793830918500064 doi: 10.1142/S1793830918500064
[23]	F. Falahati-Nezhad, M. Azari, T. Došlić, Sharp bounds on the inverse sum indeg index, Discrete Appl. Math., 217 (2017), 185–195. https://doi.org/10.1016/j.dam.2016.09.014 doi: 10.1016/j.dam.2016.09.014
[24]	I. Gutman, M. Matejić, E. Milovanović, I. Milovanović, Lower bounds for inverse sum indeg index of graphs, Kragujevac J. Math., 44 (2020), 551–562.
[25]	I. Gutman, J. M. Rodríguez, J. M. Sigarreta, Linear and non-linear inequalities on the inverse sum indeg index, Discrete Appl. Math., 258 (2019), 123–134. https://doi.org/10.1016/j.dam.2018.10.041 doi: 10.1016/j.dam.2018.10.041
[26]	M. An, L. Xiong, Some results on the inverse sum indeg index of a graph, Inf. Process. Lett., 134 (2018), 42–46. https://doi.org/10.1016/j.ipl.2018.02.006 doi: 10.1016/j.ipl.2018.02.006
[27]	J. Sedlar, D. Stevanović, A. Vasilyev, On the inverse sum indeg index, Discrete Appl. Math., 184 (2015), 202–212. https://doi.org/10.1016/j.dam.2014.11.013 doi: 10.1016/j.dam.2014.11.013
[28]	M. A. Rashid, S. Ahmad, M. K. Siddiqui, M. K. A. Kaabar, On computation and analysis of topological index-based invariants for complex coronoid systems, Complexity, 2021 (2021), 4646501. https://doi.org/10.1155/2021/4646501 doi: 10.1155/2021/4646501
[29]	M. K. Siddiqui, S. Manzoor, S. Ahmad, M. K. A. Kaabar, On computation and analysis of entropy measures for crystal structures, Math. Probl. Eng., 2021 (2021), 9936949. https://doi.org/10.1155/2021/9936949 doi: 10.1155/2021/9936949
[30]	D. A. Xavier, E. S. Varghese, A. Baby, D. Mathew, M. K. A. Kaabar, Distance based topological descriptors of zinc porphyrin dendrimer, J. Mol. Struct., 1268 (2022), 133614. https://doi.org/10.1016/j.molstruc.2022.133614 doi: 10.1016/j.molstruc.2022.133614
[31]	W. Carballosa, J. M. Rodríguez, J. M. Sigarreta, Extremal problems on the variable sum exdeg index, MATCH Commun. Math. Comput. Chem., 84 (2020), 753–772.
[32]	J. C. Hernández, J. M. Rodríguez, O. Rosario, J. M. Sigarreta, Extremal problems on the general Sombor index of graph, AIMS Math., 7 (2022), 8330–8334. https://doi.org/10.3934/math.2022464 doi: 10.3934/math.2022464
[33]	D. Vukičević, Bond additive modeling 4. QSPR and QSAR studies of the variable Adriatic indices, Croat. Chem. Acta, 84 (2011), 87–91.
[34]	R. Todeschini, P. Gramatica, E. Marengo, R. Provenzani, Weighted holistic invariant molecular descriptors. Part 2. Theory development and applications on modeling physicochemical properties of polyaromatic hydrocarbons, Chemom. Intell. Lab. Syst., 27 (1995), 221–229. https://doi.org/10.1016/0169-7439(95)80026-6 doi: 10.1016/0169-7439(95)80026-6

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