Extremal problems on exponential vertex-degree-based topological indices

José M. Sigarreta; José M. Sigarreta

doi:10.3934/mbe.2022329

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 7: 6985-6995. doi: 10.3934/mbe.2022329

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Extremal problems on exponential vertex-degree-based topological indices

José M. Sigarreta ^,

Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No.54 Col. Garita, Acalpulco Gro. 39650, Mexico

Academic Editor: Hao Wang

Received: 29 March 2022 Revised: 02 May 2022 Accepted: 06 May 2022 Published: 10 May 2022

In this work we obtain new lower and upper optimal bounds for general (exponential) indices of a graph. In the same direction, we show new inequalities involving some well-known topological indices like the generalized atom-bound connectivity index $ ABC_\alpha $ and the generalized second Zagreb index $ M_2^\alpha $. Moreover, we solve some extremal problems for their corresponding exponential indices ($ e^{ABC_\alpha} $ and $ e^{M_2^{\alpha}} $).
- general sum connectivity index,
- generalized second Zagreb index,
- topological indices,
- exponential indices
Citation: José M. Sigarreta. Extremal problems on exponential vertex-degree-based topological indices[J]. Mathematical Biosciences and Engineering, 2022, 19(7): 6985-6995. doi: 10.3934/mbe.2022329

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Abstract

In this work we obtain new lower and upper optimal bounds for general (exponential) indices of a graph. In the same direction, we show new inequalities involving some well-known topological indices like the generalized atom-bound connectivity index $ ABC_\alpha $ and the generalized second Zagreb index $ M_2^\alpha $. Moreover, we solve some extremal problems for their corresponding exponential indices ($ e^{ABC_\alpha} $ and $ e^{M_2^{\alpha}} $).

References

[1]	R. Todeschini, V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem., 64 (2010), 359–372.
[2]	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
[3]	K. C. Das, I. Gutman, B. Furtula, On atom-bond connectivity index, Chem. Phys. Lett., 511 (2011), 45–454. https://doi.org/10.1016/j.cplett.2011.06.049 doi: 10.1016/j.cplett.2011.06.049
[4]	I. Gutman, B. Furtula, Vertex-degree-based molecular structure descriptors of benzenoid systems and phenylenes, J. Serb. Chem. Soc., 77 (2012), 1031–1036. https://doi.org/10.2298/JSC111212029G doi: 10.2298/JSC111212029G
[5]	I. Gutman, B. Furtula (Eds.), Recent Results in the Theory of Randić Index, Univ. Kragujevac, Kragujevac, 2008.
[6]	X. Li, I. Gutman, Mathematical Aspects of Randić Type Molecular Structure Descriptors, Univ. Kragujevac, Kragujevac, 2006.
[7]	X. Li, Y. Shi, A survey on the Randić index, MATCH Commun. Math. Comput. Chem., 59 (2008), 127–156.
[8]	J. A. Rodríguez, J. M. Sigarreta, On the Randić index and condicional parameters of a graph, MATCH Commun. Math. Comput. Chem., 54 (2005), 403–416.
[9]	B. Borovićanin, 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 doi: 10.1016/j.amc.2016.01.017
[10]	K. C. Das, On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem., 63 (2010), 433–440.
[11]	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
[12]	B. Furtula, I. Gutman, S. Ediz, On difference of Zagreb indices, Discr. Appl. Math., 178 (2014), 83–88. https://doi.org/10.1016/j.dam.2014.06.011 doi: 10.1016/j.dam.2014.06.011
[13]	V. R. Kulli, Multiplicative Connectivity Indices of Nanostructures, LAP LEMBERT Academic Publishing, 2018. http://dx.doi.org/10.22147/jusps-A/290101
[14]	M. Liu, A simple approach to order the first Zagreb indices of connected graphs, MATCH Commun. Math. Comput. Chem., 63 (2010), 425–432.
[15]	Z. Wang, Y. Mao, K. C. Das, Y. Shang, Nordhaus-gaddum-type results for the steiner gutman index of graphs, Symmetry, 12 (2020), 1711. https://doi.org/10.3390/sym12101711 doi: 10.3390/sym12101711
[16]	H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc., 69 (1947), 17–20. https://doi.org/10.1021/ja01193a005 doi: 10.1021/ja01193a005
[17]	F. Harary, R. Z. Norman, Some properties of line digraphs, Rend. Circ. Math. Palermo, 9 (1960), 161–169. https://doi.org/10.1007/BF02854581 doi: 10.1007/BF02854581
[18]	J. Rada, Exponential vertex-degree-based topological indices and discrimination, MATCH Commun. Math. Comput. Chem., 82 (2019), 29–41.
[19]	S. Balachandran, T. Vetrik, Exponential second Zagreb index of chemical trees, Trans. Comb., 10 (2021), 97–106. https://dx.doi.org/10.22108/toc.2020.125047.1764 doi: 10.22108/toc.2020.125047.1764
[20]	R. Cruz, J. Rada, Extremal values of exponential vertex-degree-based topological indices over graphs, Kragujevac J. Math. 46 (2022), 105–113.
[21]	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.
[22]	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
[23]	K. C. Das, S. Elumalai, S. Balachandran, Open problems on the exponential vertex-degree-based topological indices of graphs, Discrete Appl. Math., 293 (2021), 38–49. https://doi.org/10.1016/j.dam.2021.01.018 doi: 10.1016/j.dam.2021.01.018
[24]	M. Eliasi, Unicyclic and bicyclic graphs with maximum exponential second Zagreb index, Discrete Appl. Math., 307 (2022), 172–179. https://doi.org/10.1016/j.dam.2021.10.019 doi: 10.1016/j.dam.2021.10.019
[25]	H. Liu, H. Chen, J. Liu, Z. Tang, Extremal trees for the exponential reduced second Zagreb index, Contrib. Math., 3 (2021), 43–52. https://doi.org/10.47443/cm.2021.0017 doi: 10.47443/cm.2021.0017
[26]	V. R. Kulli, $\delta$-Sombor index and its exponential for certain nanotubes, Ann. Pure Appl. Math., 23 (2021), 37–42. http://dx.doi.org/10.22457/apam.v23n1a06812 doi: 10.22457/apam.v23n1a06812
[27]	Y. Shang, Lower bounds for Gaussian Estrada index of graphs, Symmetry, 10 (2018), 325. https://doi.org/10.3390/sym10080325 doi: 10.3390/sym10080325
[28]	R. Cruz, J. Monsalve, J. Rada, Extremal values of vertex-degree-based topological indices of chemical trees, Appl. Math. Comp., 380 (2020), 12581. https://doi.org/10.1016/j.amc.2020.125281 doi: 10.1016/j.amc.2020.125281
[29]	Y. C. Jun, D. H. Won, S. H. Lee, D. S. Kong, S. J. Hwang, A multimetric benthic macroinvertebrate index for the assessment of stream biotic integrity in Korea, Int. J. Environ. Res. Public Health, 9 (2012), 3599–3628. https://doi.org/10.3390/ijerph9103599 doi: 10.3390/ijerph9103599
[30]	P. Lucena-Moya, I. Pardo, An invertebrate multimetric index to classify the ecological status of small coastal lagoons in the Mediterranean ecoregion (MIBIIN), Mar. Freshw. Res., 63 (2012), 801–814. https://doi.org/10.1071/MF12104 doi: 10.1071/MF12104
[31]	F. D. Malliaros, V. Megalooikonomou, C. Faloutsos, Estimating robustness in large social graphs, Knowl. Inf. Syst., 45 (2015), 645–678. https://doi.org/10.1007/s10115-014-0810-7 doi: 10.1007/s10115-014-0810-7
[32]	R. Muratov, A. Zhamangara, R. Beisenova, L. Akbayeva, K. Szoszkiewicz, S. Jusik, et al., An attempt to prepare Macrophyte Index for Rivers for assessment watercourses in Kazakhstan, Meteorol. Hydrol. Water Manag. Res. Oper. Appl., 3 (2015), 27–32. https://doi.org/10.26491/mhwm/59592 doi: 10.26491/mhwm/59592
[33]	R. Pérez-Domínguez, S. Maci, A. Courrat, M. Lepage, A. Borja, A. Uriarte, et al., Current developments on fish-based indices to assess ecological-quality status of estuaries and lagoons, Ecol. Indic., 23 (2012), 34–45. https://doi.org/10.1016/j.ecolind.2012.03.006 doi: 10.1016/j.ecolind.2012.03.006
[34]	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. https://doi.org/10.1007/s10910-019-01008-1 doi: 10.1007/s10910-019-01008-1

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