Miriam Farber

Massachusetts Institute of Technology, Cambridge, Massachusetts, United States

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Publications (6)2.51 Total impact

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    ABSTRACT: A matrix is called totally positive if every minor of it is positive. Such matrices are well studied and have numerous applications in Mathematics and Computer Science. We study how many times the value of a minor can repeat in a totally positive matrix and show interesting connections with incidence problems in combinatorial geometry. We prove that the maximum possible number of repeated $d \times d$-minors in a $d \times n$ totally-positive matrix is $O(n^{d-\frac{d}{d+1}})$. For the case $d=2$ we also show that our bound is optimal. We consider some special families of totally postive matrices to show non-trivial lower bounds on the number of repeated minors. In doing so, we arrive at a new interesting problem: How many unit-area and axis-parallel rectangles can be spanned by two points in a set of $n$ points in the plane? This problem seems to be interesting in its own right especially since it seem to have a flavor of additive combinatorics and relate to interesting incidence problems where considering only the topology of the curves involved is not enough. We prove an upper bound of $O(n^{\frac{4}{3}})$ and provide a lower bound of $n^{1+\frac{1}{O(\log\log n)}}$.
    Journal of Combinatorial Theory, Series A. 09/2013; 128.
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    ABSTRACT: We show that the maximal number of equal entries in a totally positive (resp. totally nonsingular) $n\textrm{-by-}n$ matrix is $\Theta(n^{4/3})$ (resp. $\Theta(n^{3/2}$)). Relationships with point-line incidences in the plane, Bruhat order of permutations, and $TP$ completability are also presented. We also examine the number and positionings of equal $2\textrm{-by-}2$ minors in a $2\textrm{-by-}n$ $TP$ matrix, and give a relationship between the location of equal $2\textrm{-by-}2$ minors and outerplanar graphs.
    Linear Algebra and its Applications 09/2013; 454. · 0.97 Impact Factor
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    ABSTRACT: It is known that majorization is a complete description of the relationships between the eigenvalues and diagonal entries of real symmetric matrices. However, for large subclasses of such matrices, the diagonal entries impose much greater restrictions on the eigenvalues. Motivated by previous results about Laplacian eigenvalues, we study here the additional restrictions that come from the off-diagonal sign-pattern classes of real symmetric matrices. Each class imposes additional restrictions. Several results are given for the all nonpositive and all nonnegative classes and for the third class that appears when n = 4. Complete description of the possible relationships are given in low dimensions.
    Linear Algebra and its Applications 02/2013; 438(3):1427–1445. · 0.97 Impact Factor
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    ABSTRACT: Let $NPO(k)$ be the smallest number $n$ such that the adjacency matrix of any undirected graph with $n$ vertices or more has at least $k$ nonpositive eigenvalues. We show that $NPO(k)$ is well-defined and prove that the values of $NPO(k)$ for $k=1,2,3,4,5$ are $1,3,6,10,16$ respectively. In addition, we prove that for all $k \geq 5$, $R(k,k+1) \ge NPO(k) > T_k$, in which $R(k,k+1)$ is the Ramsey number for $k$ and $k+1$, and $T_k$ is the $k^{th}$ triangular number. This implies new lower bounds for eigenvalues of Laplacian matrices: the $k$-th largest eigenvalue is bounded from below by the $NPO(k)$-th largest degree, which generalizes some prior results.
    Discrete Mathematics 08/2011; 313(13). · 0.58 Impact Factor
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    Miriam Farber, Ido Kaminer
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    ABSTRACT: In this note we give a new upper bound for the Laplacian eigenvalues of an unweighted graph. Let $G$ be a simple graph on $n$ vertices. Let $d_{m}(G)$ and $\lambda_{m+1}(G)$ be the $m$-th smallest degree of $G$ and the $m+1$-th smallest Laplacian eigenvalue of $G$ respectively. Then $ \lambda_{m+1}(G)\leq d_{m}(G)+m-1 $ for $\bar{G} \neq K_{m}+(n-m)K_1 $. We also introduce upper and lower bound for the Laplacian eigenvalues of weighted graphs, and compare it with the special case of unweighted graphs.
    06/2011;
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    ABSTRACT: Let $NPO(k)$ be the smallest number $n$ such that all the graphs of order $n$ or more have at least $k$ non-positive eigenvalues. We show that $NPO(k)$ is well-defined, and prove that the values of $NPO(k)$ for $k=1,2,3,4,5$ are $1,3,6,10,16$ respectively. In addition, we give an upper and lower bound for $NPO(k)$ for each $k$. This yields a new lower bound for the Laplacian eigenvalues: The $k$-th largest eigenvalue is bounded from below by the $NPO(k)$-th largest degree.
    01/2011;

Publication Stats

2.51 Total Impact Points

Institutions

  • 2013
    • Massachusetts Institute of Technology
      Cambridge, Massachusetts, United States
  • 2011–2013
    • Technion - Israel Institute of Technology
      H̱efa, Haifa District, Israel