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Recurrence/factorization pattern for Laplacian characteristic polynomials in DSC(1) is conjectural

Establish that, for DSC(1) trees of all generations n, the Laplacian characteristic polynomial Π_n(λ) factorizes as Π_n(λ) = −λ ∏_{i=1}^{n} π_i(λ), with π_i(λ) generated by the specified recursions (the base cases given in equation (585) and the general recursive scheme in equation (pi_rec)), thereby confirming that the recurrence/factorization pattern observed computationally for n ≤ 9 holds for all n.

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Background

For DSC(1) (trees), the authors computed Laplacian characteristic polynomials up to n = 9 and observed a structured factorization with a family of recursively defined polynomials π_i(λ).

They present explicit base polynomials and recursive definitions but do not provide a proof that this structure persists for all generations; they state this as a conjectural extension.

References

As with the adjacency matrix spectrum, to figure out the form of $\Pi_n(\lambda)$ for any finite $n$, we examine $\Pi_n(\lambda)$ using results for $n \leq 9$ directly computed by Mathematica for trees generated by the DSC$(1)$ model, see appendix~\ref{sa2}. We then identify a recurrence relation and conjecture that this pattern continues for larger $n$. We find \begin{equation} \label{Pi-factor} \Pi_n = -\lambda\prod_{i=1}n \pi_i \end{equation}

— Deterministic simplicial complexes (2507.07402 - Dorogovtsev et al., 10 Jul 2025) in Section “Laplacian spectrum of the DSC(1) model”