On the diameter and inverse degree. (Q2918572)

The inverse degree of a finite graph \(G\) is defined as \(r(G)=\sum _{v\in V(G)}\frac {1}{ \deg (v)}\), where deg\((v)\) is the degree of a vertex \(v\). \textit{P. Erdős} et al. [Congr. Numerantium 64, 121--124 (1988; Zbl 0677.05053)] proved that, if \(G\) is a connected graph of order \(n\), then the diameter of \(G\) is less than \((6r(G) + o(1)) \frac {\log n}{\log \log n}\). \textit{P. Dankelmann, H. C. Swart} and \textit{P. van den Berg} [Discrete Math. 308, No. 5--6, 670--673 (2008; Zbl 1142.05022)] improved this bound by a factor of \(2\). The authors of this paper present sharp upper bounds for trees and unicyclic graphs, which improve the results mentioned above if \(G\) is a tree.