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Contemporary Problems in Number Theory
September 1, 2011 12:45, Moscow, Steklov Mathematical Institute, Room 530 (8 Gubkina)

Telling graph properties from its largest eigenvalue

V. F. Lev

$$\max_{X,Y}\frac{e(X,Y)}{\sqrt{|X||Y|}},$$
where the maximum is taken over all pairs of (non-empty, not necessarily disjoint) subsets of the vertex set of the graph, and $e(X,Y)$ denotes the number of edges between $X$ and $Y$.