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Introduction

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The Harary Index of a Graph

Part of the book series: SpringerBriefs in Applied Sciences and Technology ((BRIEFSMATHMETH))

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Abstract

The solution of Königsberg Bridge Problem in 1736 by a great Swiss mathematician Leonhard Euler (1707–1783) gave birth to a novel subject—Graph Theory, which also made him the father of graph theory.

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References

  1. Bondy JA, Murty USR (1976) Graph theory with applications. Macmillan Press, New York

    MATH  Google Scholar 

  2. Harary F (1969) Graph theory. Addison-Wesley, Reading

    Google Scholar 

  3. Brouwer AE, Cohen AM, Neumaier A (1989) Distance-regular graphs. Springer, Berlin

    Book  MATH  Google Scholar 

  4. Biggs NL (1993) Distance-transitive graphs. Algebraic graph theory, 2nd edn. Cambridge University Press, Cambridge

    Google Scholar 

  5. Buckley F, Harary F (1990) Distance in graphs. Addison-Wesley, Redwood

    MATH  Google Scholar 

  6. Goddard W, Oellermann OR (2011) Structural analysis of complex networks. Distance in graphs. Birkhäuser/Springer, New York, pp 49–72

    Google Scholar 

  7. Janežič D, Miličević A, Nikolić S, Trinajstić N (2007) Graph theoretical matrices in chemistry. University of Kragujevac, Kragujevac

    Google Scholar 

  8. Todeschini R, Consonni V (2000) Handbook of molecular descriptors. Wiley-VCH, Weinheim, pp 497–502

    Book  Google Scholar 

  9. Todeschini R, Consonni V (2009) Molecular descriptors for chemoinformatics, vol I, vol II. Wiley-VCH, Weinheim, pp 934–938

    Google Scholar 

  10. Gutman I, Furtula B (eds) (2012) Distance in molecular graphs-theory. University of Kragujevac, Kragujevac

    Google Scholar 

  11. Gutman I, Furtula B (eds) (2012) Distance in molecular graphs-applications. University of Kragujevac, Kragujevac

    Google Scholar 

  12. Trinajstić N (1992) Chemical graph theory. CRC Press, Boca Raton

    Google Scholar 

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

    Article  Google Scholar 

  14. Hosoya H (1971) Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Bull Chem Soc Jpn 44:2332–2339

    Article  Google Scholar 

  15. Randić M (1993) Novel molecular descriptor for structure-property studies. Chem Phys Lett 211:478–483

    Article  Google Scholar 

  16. Klein DJ, Lukovits I, Gutman I (1995) On the definition of the hyper-Wiener index for cycle-containing structures. J Chem Inf Comput Sci 35:50–52

    Article  Google Scholar 

  17. Plavšić D, Nikolić S, Trinajstić N, Mihalić Z (1993) On the Harary index for the characterization of chemical graphs. J Math Chem 12:235–250

    Article  MathSciNet  Google Scholar 

  18. Ivanciuc O, Balaban TS, Balaban AT (1993) Design of topological indices. Part 4. Reciprocal distance matrix, related local vertex invariants and topological indices. J Math Chem 12:309–318

    Article  MathSciNet  Google Scholar 

  19. Mihalić Z, Trinajstić N (1992) A graph-theoretical approach to structure-property relationships. J Chem Educ 69:701–712

    Article  Google Scholar 

  20. Xu K, Das KC (2011) On Harary index of graphs. Discret Appl Math 159:1631–1640

    Article  MATH  MathSciNet  Google Scholar 

  21. Zhou B, Du Z, Trinajstić N (2008) Harary index of landscape graphs. Int J Chem Model 1:35–44

    Google Scholar 

  22. Ivanciuc O (2000) QSAR comparative study of Wiener descriptors for weighted molecular graphs. J Chem Inf Comput Sci 40:1412–1422

    Article  Google Scholar 

  23. Ivanciuc O, Ivanciuc T, Balaban AT (2000) The complementary distance matrix, a new molecular graph metric. ACH Models Chem 137:57–82

    Google Scholar 

  24. Gutman I, Furtula B, Petrović M (2009) Terminal Wiener index. J Math Chem 46:522–531

    Article  MATH  MathSciNet  Google Scholar 

  25. Székely LA, Wang H, Wu T (2011) The sum of distances between the leaves of a tree and the semi-regular property. Discret Math 311:1197–1203

    Article  MATH  Google Scholar 

  26. Gutman I (1994) A formula for the Wiener number of trees and its extension to graphs containing cycles. Graph Theory Notes NY 27:9–15

    Google Scholar 

  27. Balaban AT (1982) Highly discriminating distance based topological index. Chem Phys Lett 89:399–404

    Article  MathSciNet  Google Scholar 

  28. Balaban AT (1983) Topological indices based on topological distances in molecular graphs. Pure Appl Chem 55:199–206

    Article  Google Scholar 

  29. Dobrynin AA, Entringer R, Gutman I (2001) Wiener index of trees: theory and applications. Acta Appl Math 66:211–249

    Article  MATH  MathSciNet  Google Scholar 

  30. Dobrynin AA, Gutman I, Klavžar S, Žiget P (2002) Wiener index of hexagonal systems. Acta Appl Math 72:247–294

    Article  MATH  MathSciNet  Google Scholar 

  31. Feng L, Ilić A (2010) Zagreb, Harary and hyper-Wiener indices of graphs with a given matching number. Appl Math Lett 23:943–948

    Article  MATH  MathSciNet  Google Scholar 

  32. Furtula B, Gutman I, Tomović Ž, Vesel A, Pesek I (2002) Wiener-type topological indices of phenylenes. Indian J Chem A 41:1767–1772

    Google Scholar 

  33. Gutman I (1997) A property of the Wiener number and its modifications. Indian J Chem A 36:128–132

    Google Scholar 

  34. Gutman I, Rada J, Araujo O (2000) The Wiener index of starlike trees and a related partial order. MATCH Commun Math Comput Chem 42:145–154

    MATH  MathSciNet  Google Scholar 

  35. Liu M, Liu B (2010) Trees with the seven smallest and fifteen greatest hyper-Wiener indices. MATCH Commun Math Comput Chem 63:151–170

    MATH  MathSciNet  Google Scholar 

  36. Nikolić S, Trinajstić N, Mihalić Z (1995) The Wiener index: development and applications. Croat Chem Acta 68:105–129

    Google Scholar 

  37. Zhou B, Trinajstić N (2010) Mathematical properties of molecular descriptors based on distances. Croat Chem Acta 83:227–242

    Google Scholar 

  38. Bonchev D, Trinajstić N (1977) Information theory, distance matrix, and molecular branching. J Chem Phys 67:4517–4533

    Article  Google Scholar 

  39. Mohar B, Babić D, Trinajstić N (1993) A novel definition of the Wiener index for trees. J Chem Inf Comput Sci 33:153–154

    Article  Google Scholar 

  40. Gutman I (1994) Selected properties of the Schultz molecular topogical index. J Chem Inf Comput Sci 34:1087–1089

    Article  Google Scholar 

  41. Nikolić S, Trinajstić N, Randić M (2001) Wiener index revisited. Chem Phys Lett 333:319–321

    Article  Google Scholar 

  42. Zhou B (2010) Reverse Wiener index. In: Gutman I, Furtula B (eds) Novel molecular structure descriptors-theory and applications II. University of Kragujevac, Kragujevac, pp 193–204

    Google Scholar 

  43. Ghorbani M (2012) Computing Wiener index of chemical compounds by cut method. In: Gutman I, Furtula B (eds) Distance in molecular graphs-theory. University of Kragujevac, Kragujevac, pp 71–83

    Google Scholar 

  44. Ghorbani M, Ashrafi AR, Zousefi S (2012) Wiener index of nanotubes and nanotori. In: Gutman I, Furtula B (eds) Distance in molecular graphs-application. University of Kragujevac, Kragujevac, pp 157–166

    Google Scholar 

  45. Balaban TS, Filip PA, Ivanciuc O (1992) Computer generation of acyclic graphs based on local vertex invariants and topological indices. Derived canonical labelling and coding trees and alkanes. J Math Chem 11:79–105

    Article  Google Scholar 

  46. Lučić B, Sović I, Plavšić D, Trinajstić N (2012) Harary matrices: definitions, properties and applications. In: Gutman I, Furtula B (eds) Distance in molecular graphs-applications. University of Kragujevac, Kragujevac, pp 3–26

    Google Scholar 

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Xu, K., Das, K.C., Trinajstić, N. (2015). Introduction. In: The Harary Index of a Graph. SpringerBriefs in Applied Sciences and Technology(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45843-3_1

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