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.