Maximum determinant of (0,1) matrices with constant line sums
| dc.contributor.author | Fallat, Shaun Michael | en_US |
| dc.date.accessioned | 2024-08-13T22:16:29Z | |
| dc.date.available | 2024-08-13T22:16:29Z | |
| dc.date.copyright | 1996 | en_US |
| dc.date.issued | 1996 | |
| dc.degree.department | Department of Mathematics and Statistics | |
| dc.degree.level | Master of Science M.Sc. | en |
| dc.description.abstract | The maximum absolute value of the determinant of n x n nonsingular (0,1) matrices that have constant row and column sums k is investigated. Recently the minimum absolute value of the determinant for matrices in this class has been proved to be k gcd(n, k) for all (n,k) ≠ (4,2). However, there appears to be no such general formula for the maximum determinant. For n ≠ 4, k = 2, the maximum determinant is proved to be 2t 1f n = 3t or 3t + 2, and 2t-l 1f n = 3t + 1. Restriction to a subset of these matrices, namely those that are symmetric and have zero trace (their graphs are regular of degree k), leaves this minimum and maximum unchanged for k = 2 For this restricted class, when n ~ 7, k = n- 3, the minimum again remains unchanged and the maximum absolute value of the determinant 1s (n - 3)3Ln/4J- 1. This maximum gives a lower bound for the maximum absolute value of the determinant of the larger class, but in general this bound is not tight. Other deteminantal values and bounds for specific n and k are derived. For reference a table is given of presently known values of the minimum and maximum absolute value of the determinant of n x n nonsingular (0,1) matrices with constant row and column sums k, and of the associated restricted class. In addition to evaluation of the maximum absolute value of the determinant, matrices are exhibited that attain these maximum values. Additional relationships are shown to exist between n x n nonsingular (0,1) matrices that have constant row and column sums 2, and the associated restricted class. A localization result is established for the subdominant eigenvalues for the matrices in the restricted class. | |
| dc.format.extent | 64 pages | |
| dc.identifier.uri | https://hdl.handle.net/1828/17730 | |
| dc.rights | Available to the World Wide Web | en_US |
| dc.title | Maximum determinant of (0,1) matrices with constant line sums | en_US |
| dc.type | Thesis | en_US |
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