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 2*n* 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 *l*_{1} 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.

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* This research is supported by the Israeli Ministry of Science and the Israel Science Foundation.

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Bilu, Y., Linial, N. Lifts, Discrepancy and Nearly Optimal Spectral Gap*.
*Combinatorica* **26, **495–519 (2006). https://doi.org/10.1007/s00493-006-0029-7

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*Mathematics Subject Classification (2000):*

- 05C22
- 05C35
- 05C50
- 05C80