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On the Adjacency spectra of alternating-oriented $n$-gonal staircase digraphs

Hiroki Minamide

Abstract

For integers $n \ge 3$ and $r \ge 1$, let $Γ_{n,r}$ be the alternating-oriented digraph obtained by gluing $r$ directed $n$-cycles along a single edge in a staircase pattern, and let $A_{n,r}$ be its adjacency matrix. A canonical $n$-layer partition puts $A_{n,r}$ into an $n$-cyclic block form and isolates a cyclic product core $K_{n,r}$, so the nonzero spectrum of $A_{n,r}$ is obtained from that of $K_{n,r}$ by taking $n$th roots. We show that $K_{n,r}$ is totally nonnegative and irreducible, and hence its nonzero eigenvalues are real, positive, and simple. It follows that all nonzero eigenvalues of $A_{n,r}$ are simple and occur in $\exp(2πi/n)$-orbits, forming unions of regular $n$-gons in the complex plane. A one-step Schur complement yields a three-term recursion in $r$ for the characteristic polynomials $Φ_{n,r} \in \mathbb{Z}[x]$. This determines both the multiplicity of the eigenvalue $0$ and the number of nonzero eigenvalues, and leads to a generating function with cubic denominator. Applying a Tran-type confinement theorem gives the uniform bound $ρ(A_{n,r}) \le (27/4)^{1/n}$ and the sharp limit $\displaystyle\lim_{r \to \infty} ρ(A_{n,r}) = (27/4)^{1/n}$ for each fixed $n$. Finally, specializing at $x=1$ relates $Φ_{n,r}(1)$ to Padovan spiral numbers and yields a complete classification of rational nonzero eigenvalues.

On the Adjacency spectra of alternating-oriented $n$-gonal staircase digraphs

Abstract

For integers and , let be the alternating-oriented digraph obtained by gluing directed -cycles along a single edge in a staircase pattern, and let be its adjacency matrix. A canonical -layer partition puts into an -cyclic block form and isolates a cyclic product core , so the nonzero spectrum of is obtained from that of by taking th roots. We show that is totally nonnegative and irreducible, and hence its nonzero eigenvalues are real, positive, and simple. It follows that all nonzero eigenvalues of are simple and occur in -orbits, forming unions of regular -gons in the complex plane. A one-step Schur complement yields a three-term recursion in for the characteristic polynomials . This determines both the multiplicity of the eigenvalue and the number of nonzero eigenvalues, and leads to a generating function with cubic denominator. Applying a Tran-type confinement theorem gives the uniform bound and the sharp limit for each fixed . Finally, specializing at relates to Padovan spiral numbers and yields a complete classification of rational nonzero eigenvalues.
Paper Structure (33 sections, 25 theorems, 73 equations, 2 figures)

This paper contains 33 sections, 25 theorems, 73 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Approach and main results
  4. Organization
  5. Cyclic block form and the polygonal spectrum
  6. The digraph $\Gamma_{n,r}$
  7. Layer partition and the induced permutation
  8. An $n$-cyclic block form
  9. Cyclic product reduction and polygonal eigenvalues
  10. Total nonnegativity and simplicity of the nonzero spectrum
  11. The block factors are totally nonnegative
  12. Irreducibility
  13. Simple nonzero spectrum
  14. A recursion for $\Phi_{n,r}$ and zero/nonzero spectral counts
  15. A three-term recursion in the gluing parameter
  16. ...and 18 more sections

Key Result

Lemma 3

\newlabellem:layer-compat1 For every edge $(u,v)\in E(\Gamma_{n,r})$ we have $\ell(v)\equiv \ell(u)+1\pmod n$. Equivalently, there are no edges within a layer, and every edge goes from $V_{n,r}^{(c)}$ to $V_{n,r}^{(c+1)}$ (indices modulo $n$).

Figures (2)

  • Figure 1: Visualizations of the digraphs $\Gamma_{3,8}$ and $\Gamma_{4,6}$.
  • Figure 2: The spectrum of $\Gamma_{3,8}$ and the dashed circle of radius $(27/4)^{1/3}$ in the complex plane.

Theorems & Definitions (52)

  • Remark 1
  • Definition 2: Layer partition
  • Lemma 3: Layer increment along edges
  • Proof 1
  • Proposition 4: $n$-cyclic block form
  • Proof 2
  • Proposition 5: Cyclic product reduction
  • Proof 3
  • Remark 6: Zero multiplicity vs. the reduction exponent
  • Proposition 7: Total nonnegativity of the core
  • ...and 42 more