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- W2952277897 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 $pi:H to G$. 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 $[-2sqrt{d-1},2sqrt{d-1}]$ (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 $[-c sqrt{d log^3d}, c sqrt{d log^3d}]$ for some constant $c$. This leads to a polynomial time algorithm for constructing arbitrarily large $d$-regular graphs, with second eigenvalue $O(sqrt{d log^3 d})$. The proof uses the following lemma: Let $A$ be a real symmetric matrix such that the $l_1$ norm of each row in $A$ is at most $d$. Let $alpha = max_{x,y in {0,1}^n, supp(x)cap supp(y)=emptyset} frac {|xAy|} {||x||||y||}$. Then the spectral radius of $A$ is at most $c alpha log(d/alpha)$, for some universal constant $c$. An interesting consequence of this lemma is a converse to the Expander Mixing Lemma." @default.
- W2952277897 created "2019-06-27" @default.
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- W2952277897 date "2003-12-01" @default.
- W2952277897 modified "2023-09-27" @default.
- W2952277897 title "Constructing expander graphs by 2-lifts and discrepancy vs. spectral gap" @default.
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