Conjectured recurrence for adjacency characteristic polynomials in DSC(1)
Prove that the characteristic polynomials of the adjacency matrices in the DSC(1) model satisfy the recursion Υ_{n+1}(λ) = (Υ_n(λ) − λ \prod_{i=1}^{n−1} Υ_i(λ)) (Υ_n(λ) + λ \prod_{i=1}^{n−1} Υ_i(λ)) for all n, i.e., that the observed recurrence pattern continues for all generations.
References
To figure out the characteristic polynomials, we use Mathematica to compute $\Upsilon_n(\lambda)$ for $n \leq 9$, spot a recurrence relation, and guess that this pattern continues for larger $n$.
— Deterministic simplicial complexes
(Dorogovtsev et al., 10 Jul 2025) in Section 3.4 (Adjacency matrix spectrum of the DSC(1) model); before Eq. (Un)