NDLI: Lifts, Discrepancy and Nearly Optimal Spectral Gap*

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Lifts, Discrepancy and Nearly Optimal Spectral Gap*

Content Provider	Springer Nature Link
Author	Linial, Nathan Bilu, Yonatan
Copyright Year	2006
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 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.
Ending Page	519
Page Count	25
Starting Page	495
File Format	PDF
ISSN	02099683
e-ISSN	14396912
Journal	Combinatorica
Issue Number	5
Volume Number	26
Language	English
Publisher	Springer-Verlag
Publisher Date	2006-01-01
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Signed and weighted graphs Random graphs Extremal problems Mathematics Graphs and linear algebra (matrices, eigenvalues, etc.) Combinatorics
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Computational Mathematics

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