Spectral symmetry in conference matrices
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AbstractA conference matrix of order n is an $$n\times n$$ n × n matrix C with diagonal entries 0 and off-diagonal entries $$\pm 1$$ ± 1 satisfying $$CC^\top =(n-1)I$$ C C ⊤ = ( n - 1 ) I . If C is symmetric, then C has a symmetric spectrum $$\Sigma $$ Σ (that is, $$\Sigma =-\Sigma $$ Σ = - Σ ) and eigenvalues $$\pm \sqrt{n-1}$$ ± n - 1 . We show that many principal submatrices of C also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction.
2018 ◽
Vol 9
(10)
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pp. 1473-1476
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2017 ◽
Vol 340
(6)
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pp. 1271-1286
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