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- W3138849309 abstract "Abstract A conference matrix of order n is an $$ntimes n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> matrix C with diagonal entries 0 and off-diagonal entries $$pm 1$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> satisfying $$CC^top =(n-1)I$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>⊤</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> </mml:math> . If C is symmetric, then C has a symmetric spectrum $$Sigma $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Σ</mml:mi> </mml:math> (that is, $$Sigma =-Sigma $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>Σ</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>Σ</mml:mi> </mml:mrow> </mml:math> ) and eigenvalues $$pm sqrt{n-1}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>±</mml:mo> <mml:msqrt> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math> . 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." @default.
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- W3138849309 date "2021-03-18" @default.
- W3138849309 modified "2023-09-26" @default.
- W3138849309 title "Spectral symmetry in conference matrices" @default.
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- W3138849309 doi "https://doi.org/10.1007/s10623-021-00858-8" @default.
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