Gap sets for the spectra of cubic graphs

<p>We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 StartRoot 2 EndRoot comma 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(2 \sqrt {2},3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket negative 3 comma negative 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[-3,-2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> achieved in cubic Ramanujan graphs and line graphs are maximal. We give constraints on spectra in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket negative 3 comma 3 right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[-3,3]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are maximally gapped and construct examples which achieve these bounds. These graphs yield new instances of maximally gapped intervals. We also show that every point in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket negative 3 comma 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[-3,3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be gapped by planar cubic graphs. Our results show that the study of spectra of cubic, and even planar cubic, graphs is subtle and very rich.</p>

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