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

For integers $n \ge 3$ and $r \ge 1$, let $\Gamma_{n,r}$ 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 $\exp(2\pi i/n)$-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 $\Phi_{n,r} \in \mathbb{Z}[x]$. 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 $\rho(A_{n,r}) \le (27/4)^{1/n}$ and the sharp limit $\displaystyle\lim_{r \to \infty} \rho(A_{n,r}) = (27/4)^{1/n}$ for each fixed $n$. Finally, specializing at $x=1$ relates $\Phi_{n,r}(1)$ to Padovan spiral numbers and yields a complete classification of rational nonzero eigenvalues.