Fallat, Shaun Michael2024-08-132024-08-1319961996https://hdl.handle.net/1828/17730The 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.64 pagesAvailable to the World Wide WebThesis