The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs

2018/11/08 by Yi-Zheng Fan, Tao Huang, Yan-Hong Bao +3 · 6 citations
Mathematics · Computer Science · #Tensor decomposition and applications #Matrix Theory and Algorithms

paper · doi:10.1090/tran/7741

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a weakly irreducible nonnegative tensor with spectral radius <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho left-parenthesis script upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> ρ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">ρ (\mathcal A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (resp., <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper D Superscript left-parenthesis 0 right-parenthesis"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">D</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathfrak D⁽⁰⁾</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) be the set of normalized diagonal matrices arising from the eigenvectors of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> corresponding to the eigenvalues with modulus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho left-parenthesis script upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> ρ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">ρ (\mathcal A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (resp., the eigenvalue <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho left-parenthesis script upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> ρ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">ρ (\mathcal A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ). It is shown that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an abelian group containing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper D Superscript left-parenthesis 0 right-parenthesis"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">D</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathfrak D⁽⁰⁾</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a subgroup, which acts transitively on the set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace e Superscript bold i StartFraction 2 pi j Over script l EndFraction Baseline script upper A colon j equals 0 comma 1 comma ellipsis comma script l minus 1 right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"></mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">i</mml:mi> </mml:mrow> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi> π </mml:mi> <mml:mi>j</mml:mi> </mml:mrow> <mml:mi> ℓ </mml:mi> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mo>:</mml:mo> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo>

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• The ordering of hypertrees and unicyclic hypergraphs by the traces of A_α-tensor
• Lexicographical ordering by spectral moments of bicyclic hypergraphs
• The stabilizing index and cyclic index of coalescence and Cartesian product of uniform hypergraphs

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