On a problem of Bermond and Bollob\'as

TL;DR

This paper addresses the Degree/Diameter Problem by investigating whether, for any positive integer c, there exist graphs with maximum degree k and diameter d whose order is at least c less than the Moore bound, providing a positive answer.

Contribution

The paper provides a positive answer to Bermond and Bollobás's question, demonstrating the existence of graphs with order significantly below the Moore bound for given parameters.

Findings

Confirmed existence of graphs with order less than Moore bound by any fixed margin

Extended understanding of the bounds in the Degree/Diameter Problem

Contributed to the theoretical framework of extremal graph constructions

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 Bollob\'as concerning the Degree/Diameter Problem: Given a positive integer $c>0$, does there exist a pair $k$ and $d$, such that $n(k, d)\leq M(k,d)-c?$