Constructing expander graphs by 2-lifts and discrepancy vs. 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, √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 log³d, c√d log³ d] for some constant c. This leads to a polynomial time algorithm for constructing arbitrarily large d-regular graphs, with second eigenvalue O(√d log³ d). The proof uses the following lemma (Lemma 3.6): 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 (|xAy|)/(||x||||y||) ≤ α for every x,y ∈ {0, l}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.