Expanders obtained from affine transformations


A bipartite graphG=(U, V, E) is an (n, k, δ, α) expander if |U|=|V|=n, |E|≦kn, and for anyXU with |X|≦αn, |Γ G (X)|≧(1+δ(1−|X|/n)) |X|, whereΓ G (X) is the set of nodes inV connected to nodes inX with edges inE. We show, using relatively elementary analysis in linear algebra, that the problem of estimating the coefficientδ of a bipartite graph is reduced to that of estimating the second largest eigenvalue of a matrix related to the graph. In particular, we consider the case where the bipartite graphs are defined from affine transformations, and obtain some general results on estimating the eigenvalues of the matrix by using the discrete Fourier transform. These results are then used to estimate the expanding coefficients of bipartite graphs obtained from two-dimensional affine transformations and those obtained from one-dimensional ones.

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Jimbo, S., Maruoka, A. Expanders obtained from affine transformations. Combinatorica 7, 343–355 (1987). https://doi.org/10.1007/BF02579322

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