## Abstract

If G is a connected graph with vertex set V, then the degree distance of G, D^{′} (G), is defined as ∑_{{u, v} ⊆ V} (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. We prove the asymptotically sharp upper bound D^{′} (G) ≤ frac(1, 4) n d (n - d)^{2} + O (n^{7 / 2}) for graphs of order n and diameter d. As a corollary we obtain the bound D^{′} (G) ≤ frac(1, 27) n^{4} + O (n^{7 / 2}) for graphs of order n. This essentially proves a conjecture by Tomescu [I. Tomescu, Some extremal properties of the degree distance of a graph, Discrete Appl. Math. (98) (1999) 159-163].

Original language | English |
---|---|

Pages (from-to) | 2773-2777 |

Number of pages | 5 |

Journal | Discrete Applied Mathematics |

Volume | 157 |

Issue number | 13 |

DOIs | |

Publication status | Published - 6 Jul 2009 |

Externally published | Yes |

## Keywords

- Degree distance
- Diameter
- Distance

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics