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Ranks of random matrices and graphs
Descriptive
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Text
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Title
Ranks of random matrices and graphs
Identifier
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.15810
Identifier
ETD_470
Language
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English
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theses
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Topic
Mathematics
Subject
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Topic
Random matrices
Subject
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(authority = ETD-LCSH)
Topic
Random graphs
Subject
(ID = SBJ-1);
(authority = ETD-LCSH)
Topic
Graph theory
Abstract
Let Q_n be a random symmetric matrix whose entries on and above the main diagonal are independent random variables (e.g. the adjacency matrix of an Erdos-Renyi random graph). In this thesis we study the behavior of the rank of the matrix in terms of the probability distribution of the entries. One main result is that if the entries of the matrix are not too concentrated on any particular value, then the matrix will with high probability (probability tending to 1 as the size of the matrix tends to infinity) have full rank.
Other results concern the case where Q_n is sparse (each entry is 0 with high probability), and in particular on the adjacency matrix of sparse random graphs. In this case, we show that if the matrix is not too sparse than with high probability any dependency among the rows of Q_n will come from a dependency involving very few rows.
Other results concern the case where Q_n is sparse (each entry is 0 with high probability), and in particular on the adjacency matrix of sparse random graphs. In this case, we show that if the matrix is not too sparse than with high probability any dependency among the rows of Q_n will come from a dependency involving very few rows.
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vi, 66 pages
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Note
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Ph.D.
Note
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Includes bibliographical references (p. 64-65).
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Costello
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Kevin
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1981-
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Kevin Costello
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Kahn
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Jeffry
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Jeffry Kahn
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Michael
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Michael Saks
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Endre Boros
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Rutgers University
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Graduate School - New Brunswick
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2007
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2007
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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doi:10.7282/T30V8D7V
Genre
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ETD doctoral
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Rights
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Copyright protected
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Open
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Kevin Costello
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Rutgers University. Graduate School - New Brunswick
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Non-exclusive ETD license
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Author Agreement License
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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