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Ranks of matrices with few distinct entries | Boris Bukh
; | Date: |
1 Aug 2015 | Abstract: | An $L$-matrix is a matrix whose off-diagonal entries belong to a set $L$, and
whose diagonal is zero. Let $N(r,L)$ be the maximum size of a square $L$-matrix
of rank at most $r$. Many applications of linear algebra in extremal
combinatorics involve a bound on $N(r,L)$. We review some of these
applications, and prove several new results on $N(r,L)$. In particular, we
classify the sets $L$ for which $N(r,L)$ is linear, and show that if $N(r,L)$
is superlinear and $Lsubset mathbb{Z}$, then $N(r,L)$ is at least quadratic.
As a by-product of the work, we asymptotically determine the maximum
multiplicity of an eigenvalue $lambda$ in an adjacency matrix of a digraph of
a given size. | Source: | arXiv, 1508.0145 | Services: | Forum | Review | PDF | Favorites |
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