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Towers of Ramanujan graphs - Li, Winnie (Auteur de la Conférence) | CIRM H

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A $d$-regular graph is Ramanujan if its nontrivial eigenvalues in absolute value are bounded by $2\sqrt{d-1}$. By means of number-theoretic methods, infinite families of Ramanujan graphs were constructed by Margulis and independently by Lubotzky-Phillips-Sarnak in 1980's for $d=q+ 1$, where q is a prime power. The existence of an infinite family of Ramanujan graphs for arbitrary d has been an open question since then. Recently Adam Marcus, Daniel Spielman and Nikhil Srivastava gave a positive answer to this question by showing that any bipartite $d$-regular Ramanujan graph has a $2$-fold cover that is also Ramanujan. In this talk we shall discuss their approach and mention similarities with function field towers.[-]
A $d$-regular graph is Ramanujan if its nontrivial eigenvalues in absolute value are bounded by $2\sqrt{d-1}$. By means of number-theoretic methods, infinite families of Ramanujan graphs were constructed by Margulis and independently by Lubotzky-Phillips-Sarnak in 1980's for $d=q+ 1$, where q is a prime power. The existence of an infinite family of Ramanujan graphs for arbitrary d has been an open question since then. Recently Adam Marcus, ...[+]

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