Lifts, discrepancy and nearly optimal spectral gap

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√dlog³ d,c√dlog³ d] for some constant c. This leads to a deterministic polynomial time algorithm for constructing arbitrarily large d-regular graphs, with second eigenvalue O(√dlog³d). The proof uses the following lemma (Lemma 3.3): Let A be a real symmetric matrix with zeros on the diagonal. Let d be such that the l₁ norm of each row in A is at most d. Suppose that |x^t Ay|/∥x∥∥y∥ ≤ α for every x,y ∈ {0,1} ⁿ with 〈x,y〉=0. Then the spectral radius of A is O(α(log(d/ α)+1)). An interesting consequence of this lemma is a converse to the Expander Mixing Lemma.