The expected eigenvalue distribution of a large regular graph

“…A probabilistie model for these graphs was introduced in [10]. In [23] McKay determined the asymptotic behaviour of the eigenvalues thus providing an analogue of the semi-circle law for regular graphs. His results imply that with probability 1 --o(1) all but o(n) of the eigenvalues have absolute value <_-2 t/d -1.…”

Section: Generating Random Expanders and Magnifiersmentioning

confidence: 99%

Eigenvalues and expanders

1986

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Linear expanders have numerous applications to theoretical computer science. Here we show that a regular bipartite graph is an expander ifandonly if the second largest eigenvalue of its adjacency matrix is well separated from the first. This result, which has an analytic analogue for Riemannian manifolds enables one to generate expanders randomly and check efficiently their expanding properties. It also supplies an efficient algorithm for approximating the expanding properties of a graph. The exact determination of these properties is known to be coNP-complete.

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“…On trouve une mesure µ p différente de µ ∞ , cf. n o 2.3; cette mesure intervenait déjà dans [17] et [19],à propos des valeurs propres de certains graphes, et elle a une interprétation simple en termes de mesures de Plancherel, cf. n o 2.3.…”

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“…[16], [17]); dans le cas extrême où il y a "très peu" de circuits, on retrouve uné equirépartition suivant la mesure µ q du n o 2.3, cf. [19]. …”

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