On a problem of Bermond and Bollobás

Abstract

Let $n(k,d)$ be the order of the largest undirected graphs of maximum degree $k$ and diameter $d$, and let $M(k,d)$ be the corresponding Moore bound. In this paper, we give a positive answer to the question of Bermond and Bollobás concerning the Degree/Diameter Problem: Given a positive integer $c>0$, does there exist a pair $k$ and $d$, such that $n(k,d)\le M(k,d)-c?$