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An analogue of the Hadamard conjecture for n × n matrices with n ≡2 (mod 4)

Part of: Graph theory

Published online by Cambridge University Press:  09 April 2009

Charles H. C. Little
Charles H. C. Little
Affiliation:
Department of Mathematics and StatisticsMassey UniversityPalmerston North, New Zealand
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Abstract

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It is known that the problem of settling the existence of an n × n Hadamard matrix, where n is divisible by 4, is equivalent to that of finding the cardinality of a smallest set T of 4-circuits in the complete bipartite graph Kn, n such that T contains at least one circuit of each copy of K2,3 in Kn, n. Here we investigate the case where n ≡ 2 (mod 4), and we show that the problem of finding the cardinality of T is equivalent to that of settling the existence of a certain kind of n × n matrix. Moreover, we show that the case where n ≡ 2 (mod 4) differs from that where n ≡ 0 (mod 4) in that the problem of finding the cardinality of T is not equivalent to that of maximising the determinant of an n × n (1,-1)-matrix.

MSC classification

Secondary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 3 , June 1987 , pp. 287 - 295
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Bondy, J. A. and Murty, U. S. R., Graph theory with applications (Macmillan, London, 1976).CrossRefGoogle Scholar
[2]Brenner, J. and Cummings, L., ‘The Hadamard maximum determinant problem’, Amer. Math. Monthly 79 (1972), 626–630.CrossRefGoogle Scholar
[3]Ehlich, H., ‘Determinantenabschätzungen für binäre Matrizen’, Math. Z. 83 (1964), 123–132.CrossRefGoogle Scholar
[4]Erdös, P., ‘On the graph-theorem of Turán’ (Hungarian), Mat. Lapok 21 (1970), 249–251.Google Scholar
[5]Little, C. H. C. and Thuente, D. J., ‘The Hadamard conjecture and circuits of length four in a complete bipartite graph’, J. Austral. Math. Soc. Ser. A 31 (1981), 252–256.CrossRefGoogle Scholar
[6]Turán, P., ‘Eine Extremalaufgabe aus der Graphentheorie’, Mat. Fiz. Lapok 48 (1941), 436–452.Google Scholar
[7]Yang, C. H., ‘Some designs for maximal (+1,-1)-determinant of order n ≡ 2 (mod 4)’, Math. Comp. 20 (1966), 147–148.Google Scholar