On diameter and inverse degree of a graph

The inverse degree r(G)r(G) of a finite graph G=(V,E)G=(V,E) is defined as r(G)=∑v∈V1degv, where degv is the degree of vertex vv. 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].