Constructing expander graphs by 2-lifts and discrepancy vs. spectral gap

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We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map π:H → G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n ``new'' eigenvalues. We conjecture that every d-regular graph has a 2-lift such that all new eigenvalues are in the range [-2√d-1,2√d-1] (If true, this is tight, e.g. by the Alon-Boppana bound). Here we show that every graph of maximal degree d has a 2-lift such that all ``new'' eigenvalues are in the range [-c √d ³d, c √d ³d] for some constant c. This leads to a polynomial time algorithm for constructing arbitrarily large d-regular graphs, with second eigenvalue O(√d ³ d). The proof uses the following lemma: Let A be a real symmetric matrix such that the l₁ norm of each row in A is at most d. Let α=,y ∈ {0,1}ⁿ, supp(x)∩ supp(y)= |xAy|/||x||||y||. Then the spectral radius of A is at most c α(d/α), for some universal constant c. An interesting consequence of this lemma is a converse to the Expander Mixing Lemma.