Generalized Hadamard Matrices and Generalized Hadamard Graphs
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Published
Dec 3, 2020
Page:
65-75
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W. V. Nishadi
Department of Mathematics, Faculty of Science, University of Peradeniya, Peradeniya, Sri Lanka.
A. A. I. Perera
Department of Mathematics, Faculty of Science, University of Peradeniya, Peradeniya, Sri Lanka
Abstract
Hadamard matrices and their applications have steadily and rapidly grown during the last 2 decades. Due to that many researchers have developed various concepts on Hadamard matrices. This paper concentrates on Generalized Hadamarad matrices. In the first part of this work, some new results on construction of generalized Hadamard matrices GH(p; pn) over Cp are introduced. In the second part, graphs obtained from generalized Hadamard matrices are introduced, namely generalized Hadamard graphs. In particular, we show that the generalized Hadamard graphs are pn- regular. Our results have been illustrated by constructing pn- regular graphs for different values of p and n.
Keywords:
Generalized Hadamard matrix, Hadamard matrix, Kronecker product, regular graph, Latin square
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Original Research Article
References
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Nishadi WV, Dhananjaya KDE, Perera AAI. An algorithm to construct symmetric latin squares of order qn for q ≥ 2 and n ≥ 1. American Journal of Engineering Research. 2017;6(2):42-50.
Vangelars H, Koukouvinos C, Seberry J. Applications of Hadamard matrices. Journal of Telecommunication and Information Technology; 2003.
Brouwer AE, Cohen AM, Neumaier A. Distance regular graphs. New York: Springer-Verlag. 1989;19-20.
Ito N. Hadamard graphs. I, Graphs and Combinatorics. 1985;1:57-64.
Brock BW, Compton R, De Launey W, Seberry J. On generalized Hadamard matrices and difference matrices:Z6; 2015.
Horadam KJ. Hadamard matrices and their applications. Princeton: Princeton University Press; 2007.
Craigen R, De Launey W. Generalized Hadamard matrices whose transposes are not generalized Hadamard matrices. In: J. Combinatorial Designs. 2009;17(6):456458.
Butson, A.T. (1962). Generalized Hadamard matrices. Proc. Amer. Math. Soc, 13 , 894898.
De Launey W. Generalized Hadamard matrices which are developed modulo a group. Discrete Mathematics. 1992;104:49-65.
Colbourn CJ, Dinitz JH. Handbook of combinatorial designs (second ed.). Chapman and Hall, CRC, Boca Raton, FL. 2007;301305.
Wallis WD. Certain graphs arising from Hadamard matrices. BULL. AUSTRAL. MATH. SOC. 1969;I:325-331.
Wilson RJ. Introduction to graph theory (Fourth Ed.); 1996.
Nishadi WV, Dhananjaya KDE, Perera AAI. An algorithm to construct symmetric latin squares of order qn for q ≥ 2 and n ≥ 1. American Journal of Engineering Research. 2017;6(2):42-50.