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Contribution

Bermond/Bollobas Problem and Ramanujan Graphs

Speakers

  • Prof. Robert JAJCAY

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Content

If we denote by $n(k, d)$ 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 proven 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 a question of Bermond and Bollobas who asked whether it is true that for any a positive integer $c>0$ there exist a pair $k$ and $d$, such that $n(k, d)\leq M(k,d)-c$.

We show a surprising connection of this question to the value $2\sqrt{k-1}$, which is also essential in the definition of the Ramanujan graphs which are $k$-regular graphs having the property that their second largest eigenvalue (in modulus) does not exceed $2 \sqrt{k-1}$. We further reinforce this surprising connection by showing an interesting consequence of a negative answer to the problem of Bermond and Bollobas. Namely, we show that if there exists 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 $d$'s, the largest undirected graphs of degree $k$ and diameter $d$ must be Ramanujan graphs. Since Ramanujan graphs are scarce, our result seems to suggest a positive answer to the question of Bermond and Bollobas.

This is a joint work with Slobodan Filipovski.