Acta Applicandae Mathematica

, Volume 92, Issue 1, pp 15–20

All but 49 Numbers are Wiener Indices of Trees



The Wiener index is one of the main descriptors that correlate a chemical compound’s molecular graph with experimentally gathered data regarding the compound’s characteristics. A long standing conjecture on the Wiener index ([4, 5]) states that for any positive integer \(n\) (except numbers from a given 49 element set), one can find a tree with Wiener index \(n\). In this paper, we prove that every integer \(n>10^8\) is the Wiener index of some short caterpillar tree with at most six non-leaf vertices. The Wiener index conjecture for trees then follows from this and the computational results in [8] and [5].

Mathematics Subject Classification (2000)


Key words

trees Wiener index molecular graph 


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  1. 1.
    Dobrynin, A.A., Entringer, R., Gutman, I.: Wiener index of trees: Theory and applications. Acta Appl. Math. 66, 211–249 (2001)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Goldman, D., Istrail, S., Lancia, G., Piccolboni, A.: Algorithmic strategies in combinatorial chemistry. In: Proc. 11th ACM-SIAM Sympos. Discrete Algorithms, pp. 275–284, (2000)Google Scholar
  3. 3.
    Grosswald, E.: Representations of Integers as Sums of Squares. Springer, Berlin Heidelberg New York (1985)MATHGoogle Scholar
  4. 4.
    Gutman, I., Yeh, Y.: The sum of all distances in bipartite graphs. Math. Slovaca 45, 327–334 (1995)MATHMathSciNetGoogle Scholar
  5. 5.
    Lepović, M., Gutman, I.: A collective property of trees and chemical trees. J. Chem. Inf. Comput. Sci. 38, 823–826 (1998)CrossRefGoogle Scholar
  6. 6.
    Wiener, H.: Structural determination of paraffin boiling points. J. Amer. Chem. Soc. 69, 17–20 (1947)CrossRefGoogle Scholar
  7. 7.
    Ban, Y.A., Bespamyatnikh, S., Mustafa, N.H.: On a conjecture on Wiener indices in combinatorial chemistry. In: Proc. of the 9th International Computing and Combinatorics Conference ’03, pp. 509–518, 2003. (The journal version will appear in Algorithmica, 2004)Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA
  2. 2.Department of MathematicsUniversity of South CarolinaColumbiaUSA

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