## Abstract

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 eigen-values 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 log^{3} d, c√d log^{3} d] for some constant c. This leads to a polynomial time algorithm for constructing arbitrarily large d-regular graphs, with second eigenvalue O(√d log ^{3} 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 l _{1} norm of each row in A is at most d. Suppose that |xAy|/||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.

Original language | American English |
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Pages (from-to) | 404-412 |

Number of pages | 9 |

Journal | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

State | Published - 2004 |

Event | Proceedings - 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2004 - Rome, Italy Duration: 17 Oct 2004 → 19 Oct 2004 |

## Keywords

- Discrepancy
- Expander Graphs
- Lifts
- Lifts of Graphs
- Signed Graphs