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Spectral Characterization and Network Systems Dynamics of Zero-Divisor Topologies over $\mathbb{F}_p[x]/\langle x^4\rangle$: Consensus, Reliability, and Transport — arXiv:2609.22909

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.22909
License (as archived): CC0-1.0
Archived: 2026-10-09T21:55:23.309Z
SHA-256 of the archived text: 9126cbe9e2e7e8019bc0b83e7c6cb98c8619654cf31eed94a343ea275bdb0e30
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How to cite

Spectral Characterization and Network Systems Dynamics of Zero-Divisor Topologies over $\mathbb{F}_p[x]/\langle x^4\rangle$: Consensus, Reliability, and Transport — arXiv:2609.22909 — arXiv (en), https://arxiv.org/abs/2609.22909 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_dcd223d143ec (Metraipolis archive dc_dcd223d143ec, sha256 9126cbe9e2e7e8019bc0b83e7c6cb98c8619654cf31eed94a343ea275bdb0e30)

BibTeX

@misc{dcdcd223d143ec,
  title = {Spectral Characterization and Network Systems Dynamics of Zero-Divisor Topologies over $mathbbF_p[x]/langle x^4rangle$: Consensus, Reliability, and Transport — arXiv:2609.22909},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.22909},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_dcd223d143ec; sha256 9126cbe9e2e7e8019bc0b83e7c6cb98c8619654cf31eed94a343ea275bdb0e30},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.22909. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

For an odd prime $p$, let $\Gamma(R)$ denote the zero-divisor graph associated with the finite local ring $R= \mathbb{F}_p[x]/\langle x^4 \rangle$, having $p^3 - 1$ vertices and size $|E(\Gamma(R))| = \frac{1}{2}(3p^4 - 4p^3 - p^2 + 2)$. Using an equitable 4-cell partition induced by the ideal filtration $\langle u^3 \rangle \subset \langle u^2 \rangle \subset \langle u \rangle$, we prove that $\operatorname{Spec}_A(\Gamma(R))$ decomposes into discrete levels $\lambda \in \{0, -1\}$ with multiplicities $p^3 - p^2 - 1$ and $p^2 - 3$, respectively, alongside three irrational roots of an irreducible cubic polynomial $P_3(\lambda) = 0$, yielding adjacency energy $\epsilon(\Gamma(R)) = 4p^2 + \mathcal{O}(p)$. In contrast, the Laplacian spectrum is entirely integral: $\operatorname{Spec}_L(\Gamma(R)) = \{0^1, (p - 1)^{p^3 - p^2}, (p^2 - 1)^{p^2 - p - 1}, (p^3 - 1)^{p - 1}\}$. Consequently, the tree entropy satisfies $z(\Gamma(R)) = \ln p - \frac{1}{p} + \frac{\ln p}{p} + \mathcal{O}(\frac{\ln p}{p^2})$, with small-failure unreliability bounded by $p^2(p - 1)(1 - q)^{p-1}$. In linear consensus $\dot{x} = -Lx$, error trajectories decouple into three modal timescales $\{ (p^3 - 1)^{-1}, (p^2 - 1)^{-1}, (p - 1)^{-1} \}$ with algebraic connectivity $\alpha(\Gamma(R)) = p - 1$ and steady-state $H_2$ error variance $H_2^2 = \frac{Kf(\Gamma(R))}{2n^2} = \frac{1}{2p} + \mathcal{O}(p^{-3})$. Furthermore, the Kirchhoff index yields a vanishing average resistance distance $\overline{r}(\Gamma(R)) = \frac{2}{p} + \mathcal{O}(p^{-3}) \to 0$, while unbiased Markovian random walks induce an asymptotic traffic concentration vector $\Pi = (\frac{1}{3}, 0, \frac{1}{3}, \frac{1}{3})$ with quadratic per-node core routing stress $\pi_{\mathrm{core}}/\pi_{\mathrm{periph}} \sim p^2 + p + 1$.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-22 in math.IT).
Categorie arXiv: Information Theory (cs.IT).
Pagina del lavoro: https://arxiv.org/abs/2609.22909 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-09T21:44:07.457Z dalla pagina https://arxiv.org/catchup/math.IT/2026-09-22?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).