Closed walks and eigenvalues of Abelian Cayley graphs
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Abstract
We show that Abelian Cayley graphs contain many closed walks of even length. This implies that given <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>k</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>3</mml:mn> </mml:math> , for each <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">ϵ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> , there exists <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>C</mml:mi> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">ϵ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> such that for each Abelian group G and each symmetric subset S of G with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo>∉</mml:mo> <mml:mi>S</mml:mi> </mml:math> , the number of eigenvalues <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> of the Cayley graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>⩾</mml:mo> <mml:mi>k</mml:mi> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">ϵ</mml:mi> </mml:math> is at least <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>C</mml:mi> <mml:mo>⋅</mml:mo> <mml:mo stretchy="false">|</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">|</mml:mo> </mml:math> . This can be regarded as an analogue for Abelian Cayley graphs of a theorem of Serre for regular graphs.
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