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 ${\left[ { - 2{\sqrt {d - 1} },2{\sqrt {d - 1} }} \right]}$ (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 ${\left[ { - c{\sqrt {d\log ^{3} d} },c{\sqrt {d\log ^{3} d} }} \right]}$ for some constant c. This leads to a deterministic polynomial time algorithm for constructing arbitrarily large d-regular graphs, with second eigenvalue $O{\left( {{\sqrt {d\log ^{3} d} }} \right)}$.

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 $\frac{{{\left| {x^{t} Ay} \right|}}} {{{\left\| x \right\|}{\left\| y \right\|}}} \leqslant \alpha$ 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.