Diameter and inverse degree

The inverse degree r(G)r(G) of a finite graph G=(V,E)G=(V,E) is defined as r(G)=∑v∈V1degv. We prove that, if G is connected and of order n, then the diameter of G   is less than (3r(G)+2+o(1))lognloglogn. This improves a bound given by Erdös et al. by a factor of approximately 2.