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WIENER INDEX OF TREES OF GIVEN ORDER AND DIAMETER AT MOST $6$

Published online by Cambridge University Press:  19 September 2013

SIMON MUKWEMBI
Affiliation:
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa email [email protected]
TOMÁŠ VETRÍK*
Affiliation:
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa
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Abstract

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The long-standing open problem of finding an upper bound for the Wiener index of a graph in terms of its order and diameter is addressed. Sharp upper bounds are presented for the Wiener index, and the related degree distance and Gutman index, for trees of order $n$ and diameter at most $6$.

Type
Research Article
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Andova, V., Dimitrov, D., Fink, J. and Skrekovski, R., ‘Bounds on Gutman index’, MATCH Commun. Math. Comput. Chem. 67 (2012), 515524.Google Scholar
Bucicovschi, O. and Cioabǎ, S. M., ‘The minimum degree distance of graphs of given order and size’, Discrete Appl. Math. 156 (2008), 35183521.CrossRefGoogle Scholar
Dankelmann, P., Gutman, I., Mukwembi, S. and Swart, H. C., ‘On the degree distance of a graph’, Discrete Appl. Math. 157 (2009), 27732777.CrossRefGoogle Scholar
Dankelmann, P., Gutman, I., Mukwembi, S. and Swart, H. C., ‘The edge-Wiener index of a graph’, Discrete Math. 309 (2009), 34523457.Google Scholar
DeLaViña, E. and Waller, B., ‘Spanning trees with many leaves and average distance’, Electron. J. Combin. 15 (2008), 116.CrossRefGoogle Scholar
Gutman, I., ‘Selected properties of the Schultz molecular topological index’, J. Chem. Inf. Comput. Sci. 34 (1994), 10871089.Google Scholar
Klein, D. J., Mihalić, Z., Plavšić, D. and Trimajstrić, N., ‘Molecular topological index, a relation with the Wiener index’, J. Chem. Inf. Comput. Sci. 32 (1992), 304305.CrossRefGoogle Scholar
Morgan, M. J., Mukwembi, S. and Swart, H. C., ‘On the eccentric connectivity index of a graph’, Discrete Math. 311 (2011), 12291234.Google Scholar
Mukwembi, S., ‘On the upper bound of Gutman index of graphs’, MATCH Commun. Math. Comput. Chem. 68 (2012), 343348.Google Scholar
Plesník, J., ‘Critical graph of given diameter’, Acta Math. Univ. Comenian. (N.S.) 30 (1975), 7193.Google Scholar
Tomescu, I., ‘Some extremal properties of the degree distance of a graph’, Discrete Appl. Math. 98 (1999), 159163.CrossRefGoogle Scholar
Zhou, T., Xu, J. and Liu, J., ‘On diameter and average distance of graphs’, OR Trans. 8 (2004), 16.Google Scholar