A Connection Between a Question of Bermond and Bollobás and Ramanujan Graphs

If we let $n(k, d)$ denote the order of the largest undirected graphs of maximum degree $k$ and diameter $d$ , and let $M(k,d)$ denote the corresponding Moore bound, then $n(k,d) \leq M(k,d)$ , for all $k \geq 3$ , $d \geq 2 $ . While the inequality has been proved strict for all but very few pairs $k$ and $d$ , the exact relation between the values $n(k,d)$ and $M(k,d)$ is unknown, and the uncertainty of the situation is captured by an open question of Bermond and Bollobas who asked whether it is true that for any positive integer $c>0$ there exist a pair $k$ and $d$ , such that $n(k, d)\leq M(k,d)-c$ . We present a connection of this question to the value $2\sqrt{k-1}$ , which is also essential in the definition of the Ramanujan graphs defined as $k$ -regular graphs whose second largest eigenvalue (in modulus) does not exceed $2 \sqrt{k-1}$ . We further reinforce this surprising connection by showing that if the answer to the question of Bermond and Bollobas were negative and there existed a $c > 0$ such that $n(k,d) \geq M(k,d) - c $ , for all $k \geq 3$ , $d \geq 2 $ , then, for any fixed $k$ and all sufficiently large even $d$ ’s, the largest undirected graphs of degree $k$ and diameter $d$ would have to be Ramanujan graphs. This would imply a positive answer to the open question whether infinitely many non-bipartite $k$ -regular Ramanujan graphs exist for any degree $k$ .