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New lower bounds for the second variable Zagreb index

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Abstract

The aim of this paper is to obtain new sharp inequalities for a large family of topological indices, including the second variable Zagreb index \(M_2^{\alpha }\), and to characterize the set of extremal graphs with respect to them. Our main results provide lower bounds on this family of topological indices involving just the minimum and the maximum degree of the graph. These inequalities are new even for the Randić, the second Zagreb and the modified Zagreb indices.

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References

  • Andova V, Petrusevski M (2011) Variable Zagreb indices and Karamata’s inequality. MATCH Commun Math Comput Chem 65:685–690

    MathSciNet  MATH  Google Scholar 

  • Drmota M (2009) Random trees. An interplay between combinatorics and probability. Springer, Wien-New York

    MATH  Google Scholar 

  • Fajtlowicz S (1987) On conjectures of Graffiti-II. Congr Numer 60:187–197

    MathSciNet  MATH  Google Scholar 

  • Ghorbani M, Songhori M, Gutman I (2012) Modified Narumi–Katayama index. Kragujevac J Sci 34:57–64

    Google Scholar 

  • Gutman I (2013) Degree-based topological indices. Croat Chem Acta 86:351–361

    Article  Google Scholar 

  • Gutman I, Das KC (2004) The first Zagreb index 30 years after. MATCH Commun Math Comput Chem 50:83–92

    MathSciNet  MATH  Google Scholar 

  • Gutman I, Réti T (2014) Zagreb group indices and beyond. Int J Chem Model 6(2–3):191–200

    Google Scholar 

  • Gutman I, Tošović J (2013) Testing the quality of molecular structure descriptors. Vertex-degreebased topological indices. J Serb Chem Soc 78(6):805–810

    Article  Google Scholar 

  • Gutman I, Trinajstić N (1972) Graph theory and molecular orbitals. Total \(\pi \)-electron energy of alternant hydrocarbons. Chem Phys Lett 17:535–538

    Article  Google Scholar 

  • Kulli VR (2016) Multiplicative connectivity indices of certain nanotubes. Ann Pure Appl Math 12(2):169–176

    Article  Google Scholar 

  • Li X, Zhao H (2004) Trees with the first smallest and largest generalized topological indices. MATCH Commun Math Comput Chem 50:57–62

    MathSciNet  MATH  Google Scholar 

  • Li X, Zheng J (2005) A unified approach to the extremal trees for different indices. MATCH Commun Math Comput Chem 54:195–208

    MathSciNet  MATH  Google Scholar 

  • Liu M, Liu B (2010) Some properties of the first general Zagreb index. Australas J Combin 47:285–294

    MathSciNet  MATH  Google Scholar 

  • Martínez-Pérez A, Rodríguez JM (2018) New lower bounds for the geometric-arithmetic index. MATCH Commun Math Comput Chem 79(2):451–466

    MathSciNet  Google Scholar 

  • Miličević A, Nikolić S (2004) On variable Zagreb indices. Croat Chem Acta 77:97–101

    Google Scholar 

  • Miličević A, Nikolić S, Trinajstić N (2004) On reformulated Zagreb indices. Mol Divers 8:393–399

    Article  Google Scholar 

  • Milovanović EI, Milovanović IŽ, Dolićanin EĆ, Glogić E (2016) A note on the first reformulated Zagreb index. Appl Math Comput 273:16–20

    MathSciNet  Google Scholar 

  • Nikolić S, Kovačević G, Miličević A, Trinajstić N (2003) The Zagreb indices 30 years after. Croat Chem Acta 76:113–124

    Google Scholar 

  • Nikolić S, Miličević A, Trinajstić N, Jurić A (2004) On use of the variable Zagreb \(^\nu M_2\) index in QSPR: boiling points of Benzenoid hydrocarbons. Molecules 9:1208–1221

    Article  Google Scholar 

  • Randić M (1975) On characterization of molecular branching. J Am Chem Soc 97:6609–6615

    Article  Google Scholar 

  • Randić M (1991a) Novel graph theoretical approach to heteroatoms in QSAR. Chemometrics Intel Lab Syst 10:213–227

  • Randić M (1991b) On computation of optimal parameters for multivariate analysis of structure-property relationship. J Chem Inf Comput Sci 31:970–980

  • Randić M, Plavšić D, Lerš N (2001) Variable connectivity index for cycle-containing structures. J Chem Inf Comput Sci 41:657–662

    Article  Google Scholar 

  • Ranjini PS, Lokesha V, Usha A (2013) Relation between phenylene and hexagonal squeeze using harmonic index. Int J Appl Graph Theory 1:116–121

    Google Scholar 

  • Rodríguez JM, Sigarreta JM (2017) New results on the harmonic index and its generalizations. MATCH Commun Math Comput Chem 78(2):387–404

    MathSciNet  Google Scholar 

  • Sigarreta JM (2015) Bounds for the geometric–arithmetic index of a graph. Miskolc Math Notes 16:1199–1212

    Article  MathSciNet  MATH  Google Scholar 

  • Singh M, Ch. Das K, Gupta S, Madan AK (2014) Refined variable Zagreb indices: highly discriminating topological descriptors for QSAR/QSPR. Int J Chem Model 6(2–3):403–428

  • Vukičević D (2010) Bond additive modeling 2. Mathematical properties of max–min rodeg index. Croat Chem Acta 83(3):261–273

    Google Scholar 

  • Wiener H (1947) Structural determination of paraffin boiling points. J Am Chem Soc 69:17–20

    Article  Google Scholar 

  • Zhang H, Zhang S (2006) Unicyclic graphs with the first three smallest and largest values of the first general Zagreb index. MATCH Commun Math Comput Chem 55:427–438

    MathSciNet  MATH  Google Scholar 

  • Zhang S, Wang W, Cheng TCE (2006) Bicyclic graphs with the first three smallest and largest values of the first general Zagreb index. MATCH Commun Math Comput Chem 55:579–592

    MathSciNet  MATH  Google Scholar 

  • Zhou B, Trinajstić N (2010) On general sum-connectivity index. J Math Chem 47:210–218

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Facultad CC. Sociales de Talavera, Avda. Real Fábrica de Seda, s/n., 45600, Talavera de la Reina, Toledo, Spain

    Álvaro Martínez-Pérez

  2. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Madrid, Spain

    José M. Rodríguez

Authors
  1. Álvaro Martínez-Pérez
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  2. José M. Rodríguez
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Corresponding author

Correspondence to Álvaro Martínez-Pérez.

Additional information

The first author was partially supported by a Grant from Ministerio de Economía y Competititvidad (MTM 2015-63612P), Spain, the second author by two Grants from Ministerio de Economía y Competititvidad (MTM 2016-78227-C2-1-P and MTM 2015-69323-REDT), Spain, and a Grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México.

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Martínez-Pérez, Á., Rodríguez, J.M. New lower bounds for the second variable Zagreb index. J Comb Optim 36, 194–210 (2018). https://doi.org/10.1007/s10878-018-0293-7

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  • DOI: https://doi.org/10.1007/s10878-018-0293-7

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