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A Connection Between a Question of Bermond and Bollobás and Ramanujan Graphs
Acta Applicandae Mathematicae ( IF 1.6 ) Pub Date : 2021-08-24 , DOI: 10.1007/s10440-021-00429-y
Slobodan Filipovski 1 , Robert Jajcay 1, 2
Affiliation  

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 Bollobás 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 Bollobás 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 \(k\).



中文翻译:

Bermond 和 Bollobás 问题与拉马努金图之间的联系

如果我们让\(n(k, d)\)表示最大度数\(k\)和直径\(d\)的最大无向图的顺序,让\(M(k,d)\)表示相应的摩尔界限,然后\(n(k,d) \leq M(k,d)\),对于所有\(k \geq 3\)\(d \geq 2 \)。虽然除了极少数对\(k\)\(d\) 之外的所有不等式都被证明是严格的,但值\(n(k,d)\)\(M(k,d)之间的确切关系\)是未知的,并且情况的不确定性被 Bermond 和 Bollobás 的一个公开问题所捕获,他们询问对于任何正整数是否为真\(c>0\)存在一对\(k\)\(d\),使得\(n(k, d)\leq M(k,d)-c\)

我们提出了这个问题与值\(2\sqrt{k-1}\) 的联系,这在定义为\(k\) -正则图的拉马努金图的定义中也是必不可少的,其第二大特征值(在模数) 不超过\(2 \sqrt{k-1}\)。我们通过证明如果 Bermond 和 Bollobás 问题的答案是否定的并且存在\(c > 0\)使得\(n(k,d) \geq M(k,d) - c \),对于所有\(k \geq 3\)\(d \geq 2 \),然后,对于任何固定的\(k\)和所有足够大的甚至\(d\)的,最大的度数\(k\) 的无向图直径\(d\)必须是拉马努金图。这将意味着对任何程度\(k\)是否存在无限多个非二部\(k\) -正则拉马努金图的开放性问题的肯定答案。

更新日期:2021-08-25
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