On diameter and inverse degree of a graph

The inverse degree r ( G ) of a finite graph G = ( V , E ) is defined as r ( G ) = ∑ v ∈ V 1 deg v , where deg v is the degree of vertex v . We establish inequalities concerning the sum of the diameter and the inverse degree of a graph which for the most part are tight. We also find upper bounds on the diameter of a graph in terms of its inverse degree for several important classes of graphs. For these classes, our results improve bounds by Erdős et al. (1988) [5], and by Dankelmann et al. (2008) [4].