×

zbMATH — the first resource for mathematics

All but 49 numbers are Wiener indices of trees. (English) Zbl 1101.05027
Summary: 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 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 [Y. A. Ban et al., On a conjecture on Wiener indices in combinatorial chemistry. In: Proc. of the 9th International Computing and Combinatorics Conference ’03, 509–518 (2003) and Algorithmica 40, 99–117 (2004; Zbl 1088.05503)] and [M. Lepović and I. Gutman, A collective property of trees and chemical trees. J. Chem. Inf. Comput. Sci. 38, 823–826 (1998)].

MSC:
05C05 Trees
Software:
OCOTILLO
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Dobrynin, A.A., Entringer, R., Gutman, I.: Wiener index of trees: Theory and applications. Acta Appl. Math. 66, 211–249 (2001) · Zbl 0982.05044 · doi:10.1023/A:1010767517079
[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) · Zbl 0963.92015
[3] Grosswald, E.: Representations of Integers as Sums of Squares. Springer, Berlin Heidelberg New York (1985) · Zbl 0574.10045
[4] Gutman, I., Yeh, Y.: The sum of all distances in bipartite graphs. Math. Slovaca 45, 327–334 (1995) · Zbl 0853.05032
[5] Lepović, M., Gutman, I.: A collective property of trees and chemical trees. J. Chem. Inf. Comput. Sci. 38, 823–826 (1998)
[6] Wiener, H.: Structural determination of paraffin boiling points. J. Amer. Chem. Soc. 69, 17–20 (1947) · doi:10.1021/ja01193a005
[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) · Zbl 1276.92102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.