On the Diameter and Inverse Degree of a Graph

Abstract

The inverse degree \(r(G)\) of a finite graph \(G = (V, E)\) is defined by \(r(G) = \sum_{v\in V} \frac{1}{deg(v)}\) where \(deg(v)\) is the degree of \(v\) in \(G\). Erdős \(et\) \(al\). proved that, if \(G\) is a connected graph of order \(n\), then the diameter of \(G\) is less than \((6r(G) + \sigma(1))\frac{\log n}{\log \log n}\). Dankelmann et al. improved this bound by a factor of approximately \(2\). We give the sharp upper bounds for trees and unicyclic graphs, which improves the above upper bounds.