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signing
Signing a graph to have small magnitude eigenvalues ★★
Conjecture If
is the adjacency matrix of a
-regular graph, then there is a symmetric signing of
(i.e. replace some
entries by
) so that the resulting matrix has all eigenvalues of magnitude at most
.
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$ +1 $](/files/tex/155fca3c17d66548c323f203be786f9387842fe4.png)
![$ -1 $](/files/tex/26833acbe5abb13c40595cebdee81f595c59a397.png)
![$ 2 \sqrt{d-1} $](/files/tex/40581c05b66d632e7bdb2bcb852e63443663853a.png)
Keywords: eigenvalue; expander; Ramanujan graph; signed graph; signing
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