Acta Mathematica Sinica, English Series

, Volume 33, Issue 11, pp 1477–1503 | Cite as

On a classical theorem on the diameter and minimum degree of a graph

  • Verónica Hernández
  • Domingo Pestana
  • José M. Rodríguez
Article
  • 37 Downloads

Abstract

In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdös, Pach, Pollack and Tuza. We use these bounds in order to study hyperbolic graphs (in the Gromov sense). To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H(n, δ0) be the set of graphs G with n vertices and minimum degree δ0, and J(n, Δ) be the set of graphs G with n vertices and maximum degree Δ. We study the four following extremal problems on graphs: a(n, δ0) = min{δ(G) | GH(n, δ0)}, b(n, δ0) = max{δ(G) | GH(n, δ0)}, α(n, Δ) = min{δ(G) | GJ(n, Δ)} and β(n, Δ) = max{δ(G) | GJ(n, Δ)}. In particular, we obtain bounds for b(n, δ0) and we compute the precise value of a(n, δ0), α(n, Δ) and β(n, Δ) for all values of n, δ0 and Δ, respectively.

Keywords

Extremal problems on graphs diameter minimum degree maximum degree Gromov hyperbolicity hyperbolicity constant finite graphs 

MR(2010) Subject Classification

05C75 05C12 05A20 05C80 

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Verónica Hernández
    • 1
  • Domingo Pestana
    • 1
  • José M. Rodríguez
    • 1
  1. 1.Universidad Carlos III de MadridLeganésSpain

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