Expanders obtained from affine transformations

A bipartite graph G=(U, V, E) is an (n, k, δ, α) expander if |U|=|V|=n, |E|≦kn, and for any X⊆U with |X|≦αn, |Γ _G (X)|≧(1+δ(1-|X|/n)) |X|, where Γ _G (X) is the set of nodes in V connected to nodes in X with edges in E. We show, using relatively elementary analysis in linear algebra, that the problem of estimating the coefficient δ 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.