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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: mukwembi@ukzn.ac.za)
S. MUNYIRA
Affiliation:
Department of Mathematics, University of Zimbabwe, Zimbabwe
*
✉For correspondence; e-mail: mukwembi@ukzn.ac.za
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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.

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