On a Classical Theorem on the Diameter and Minimum Degree of a Graph

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 Erd¨os, 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） | G ∈ H（n, δ0）}, b（n, δ0） = max{δ（G） |G ∈ H（n, δ0）}, α（n, Δ） = min{δ（G） | G ∈ J（n, Δ）} and β（n, Δ） = max{δ（G） | G ∈ J（n, Δ）}. In particular, we obtain bounds for b（n, δ0） and we compute the precise value of a（n, δ0）, α（n, Δ） andβ（n, Δ） for all values of n, δ0 and Δ, respectively.