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DEGREE DISTANCE AND MINIMUM DEGREE

Part of: Graph theory

Published online by Cambridge University Press:  09 July 2012

S. MUKWEMBI  and
S. MUNYIRA
S. MUKWEMBI*
Affiliation:
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, South Africa (email: [email protected])
S. MUNYIRA
Affiliation:
Department of Mathematics, University of Zimbabwe, Zimbabwe
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Let G be a finite connected graph of order n, minimum degree δ and diameter d. The degree distance D′(G) of G is defined as ∑ {u,v}⊆V (G)(deg u+deg v) d(u,v), where deg w is the degree of vertex w and d(u,v) denotes the distance between u and v. In this paper, we find an asymptotically sharp upper bound on the degree distance in terms of order, minimum degree and diameter. In particular, we prove that

\[ D^\prime (G)\le \frac {1}{4}\,dn\biggl (n-\frac {d}{3}(\delta +1)\biggr )^2+O(n^3). \]
As a corollary, we obtain the bound D′ (G)≤n4 /(9(δ+1) )+O(n3) for a graph G of order n and minimum degree δ. This result, apart from improving on a result of Dankelmann et al. [‘On the degree distance of a graph’, Discrete Appl. Math.157 (2009), 2773–2777] for graphs of given order and minimum degree, completely settles a conjecture of Tomescu [‘Some extremal properties of the degree distance of a graph’, Discrete Appl. Math.98(1999), 159–163].

Keywords

degree distanceminimum degreediameter

MSC classification

Secondary: 05C12: Distance in graphs
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 2 , April 2013 , pp. 255 - 271
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc.

References

[1]Bucicovschi, O. & Cioabă, S. M., ‘The minimum degree distance of graphs of given order and size’, Discrete Appl. Math. 156 (2008), 3518–3521.CrossRefGoogle Scholar
[2]Dankelmann, P. & Entringer, R. C., ‘Average distance, minimum degree and spanning trees’, J. Graph Theory 33 (2000), 1–13.3.0.CO;2-L>CrossRefGoogle Scholar
[3]Dankelmann, P., Gutman, I., Mukwembi, S. & Swart, H. C., ‘On the degree distance of a graph’, Discrete Appl. Math. 157 (2009), 2773–2777.CrossRefGoogle Scholar
[4]Dankelmann, P. & Mukwembi, S., ‘The distance concept and distance in graphs’, in: Distance in Molecular Graphs – Theory (eds. Furtula, B. & Gutman, I.) (University of Kragujevac, Kragujevac, 2012), pp. 2–48.Google Scholar
[5]Dobrynin, A. A. & Kochetova, A. A., ‘Degree distance of a graph: a degree analogue of the Wiener index’, J. Chem. Inf. Comput. Sci. 34 (1994), 1082–1086.CrossRefGoogle Scholar
[6]Gutman, I., ‘Selected properties of the Schultz molecular topological index’, J. Chem. Inf. Comput. Sci. 34 (1994), 1087–1089.CrossRefGoogle Scholar
[7]Hou, Y. & Chang, A., ‘The unicyclic graphs with maximum degree distance’, J. Math. Study 39 (2006), 18–24.Google Scholar
[8]Ilić, A., Stevanović, D., Feng, L., Yu, G. & Dankelmann, P., ‘Degree distance of unicyclic and bicyclic graphs’, Discrete Appl. Math. 159 (2011), 779–788.CrossRefGoogle Scholar
[9]Klein, D. J., Mihalić, Z., Plavšić, D. & Trinajstić, N., ‘Molecular topological index, a relation with the Wiener index’, J. Chem. Inf. Comput. Sci. 32 (1992), 304–305.CrossRefGoogle Scholar
[10]Kouider, M. & Winkler, P., ‘Mean distance and minimum degree’, J. Graph Theory 25 (1997), 95–99.3.0.CO;2-D>CrossRefGoogle Scholar
[11]Mihalić, Z., Nikolić, S. & Trinajstić, N., ‘Comparative study of molecular descriptors derived from the distance matrix’, J. Chem. Inf. Comput. Sci. 32 (1992), 28–37.CrossRefGoogle Scholar
[12]Schultz, H. P., Schultz, E. B. & Schultz, T. P, ‘Topological organic chemistry 7. Graph theory and molecular topological indices of unsaturated and aromatic hydrocarbons’, J. Chem. Inf. Comput. Sci. 33 (1993), 863–867.CrossRefGoogle Scholar
[13]Tomescu, I., ‘Some extremal properties of the degree distance of a graph’, Discrete Appl. Math. 98 (1999), 159–163.CrossRefGoogle Scholar
[14]Tomescu, A. I., ‘Unicyclic and bicyclic graphs having minimum degree distance’, Discrete Appl. Math. 156 (2008), 125–130.CrossRefGoogle Scholar
[15]Wiener, H., ‘Structural determination of the paraffin boiling points’, J. Amer. Chem. Soc. 69 (1947), 17–20.CrossRefGoogle ScholarPubMed