Probabilistic and extremal behavior in graphs and matrices
Name
1191267375-MIT.pdf
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920.73 KB
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Adobe PDF
Checksum (MD5)
84de8bc0c3e09624228b9b11b15d2b85
Author(s)
McKinley, Gweneth(Gweneth Ann)
Advisor(s)
Henry Cohn.
Date Issued
2020
Publisher
Massachusetts Institute of Technology
Abstract
This thesis deals with several related questions in probabilistic and extremal graph theory and discrete random matrix theory. First, for any bipartite graph H containing a cycle, we prove an upper bound of [mathematical equation] on the number of labeled H-free graphs on n vertices, given only a fairly natural assumption on the growth rate of. Bounds of the form [mathematical equation] have been proven only for relatively few special graphs H, often with considerable difficulty, and our result unifies all previously known special cases. Next, we give a variety of bounds on the clique numbers of random graphs arising from the theory of graphons. A graphon is a symmetric measurable function [mathematical equation], and each graphon gives rise naturally to a random graph distribution, denoted G(n, W ), that can be viewed as a generalization of the Erdős-Ré́nyi random graph.
Recently, Doležal, Hladký, and Máthé gave an asymptotic formula of order log n for the clique number of G(n, W ) when W is bounded away from 0 and 1. We show that if W is allowed to approach 1 at a finite number of points, and displays a moderate rate of growth near these points, then the clique number of G(n, W) will be [theta]([square root n]) almost surely. We also give a family of examples with clique number [theta](n[superscript alpha]) for any [alpha] [element symbol] (0, 1) , and some conditions under which the clique number of G(n, W ) will be [omicron]([square root]n), [lower case omega]([square root]n), or [upper case omega]([superscript alpha]) for [alpha] [element symbol] (0, 1). Finally, for an nxm matrix M of independent Rademacher (±1) random variables, it is well known that if n
More precisely, if m >/- n + n[superscript 1-[epsilon]/6], then even after changing the sign of (1 - [epsilon])m/2 entries, M is still of full rank with high probability. This is asymptotically best possible, as one can easily make any two rows proportional with at most m/2 changes. Moreover, this theorem gives an asymptotic solution to a slightly weakened version of a conjecture made by Van Vu in [Vu08].
Description
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020
Cataloged from the official PDF of thesis.
Includes bibliographical references (pages 77-82).
Subjects
Mathematics.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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