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7008 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 63 of 351

  • Lower bound on the size of a quasirandom forcing set of permutations

  • Part of
  • Martin Kurečka
  • Journal:
    Combinatorics, Probability and Computing / Volume 31 / Issue 2 / March 2022
    Published online by Cambridge University Press:
    27 July 2021, pp. 304-319
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  • A set S of permutations is forcing if for any sequence $\{\Pi_i\}_{i \in \mathbb{N}}$ of permutations where the density $d(\pi,\Pi_i)$ converges to $\frac{1}{|\pi|!}$ for every permutation $\pi \in S$, it holds that $\{\Pi_i\}_{i \in \mathbb{N}}$ is quasirandom. Graham asked whether there exists an integer k such that the set of all permutations of order k is forcing; this has been shown to be true for any $k\ge 4$. In particular, the set of all 24 permutations of order 4 is forcing. We provide the first non-trivial lower bound on the size of a forcing set of permutations: every forcing set of permutations (with arbitrary orders) contains at least four permutations.

  • Spectral gap in random bipartite biregular graphs and applications

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  • Gerandy Brito, Ioana Dumitriu, Kameron Decker Harris
  • Journal:
    Combinatorics, Probability and Computing / Volume 31 / Issue 2 / March 2022
    Published online by Cambridge University Press:
    23 July 2021, pp. 229-267
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  • We prove an analogue of Alon’s spectral gap conjecture for random bipartite, biregular graphs. We use the Ihara–Bass formula to connect the non-backtracking spectrum to that of the adjacency matrix, employing the moment method to show there exists a spectral gap for the non-backtracking matrix. A by-product of our main theorem is that random rectangular zero-one matrices with fixed row and column sums are full rank with high probability. Finally, we illustrate applications to community detection, coding theory, and deterministic matrix completion.

Page 63 of 351