Spectral Characterization and Network Systems Dynamics of Zero-Divisor Topologies over 𝔽_p[x]/⟨ x⁴⟩: Consensus, Reliability, and Transport

@misc{sarkar2026spectral,
  title = {{Spectral Characterization and Network Systems Dynamics of Zero-Divisor Topologies over $\mathbbF_p[x]/\langle x^4\rangle$: Consensus, Reliability, and Transport}},
  author = {Apurba Sarkar and Kalyan Hansda and Makhan Maji},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22909},
  eprint = {2609.22909},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22909},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Spectral Characterization and Network Systems Dynamics of Zero-Divisor Topologies over 𝔽_p[x]/⟨ x^4⟩: Consensus, Reliability, and Transport
AU  - Apurba Sarkar
AU  - Kalyan Hansda
AU  - Makhan Maji
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.22909
UR  - https://arxiv.org/abs/2609.22909
AB  - For an odd prime p, let Γ(R) denote the zero-divisor graph associated with the finite local ring R= 𝔽_p[x]/⟨ x^4 ⟩, having p^3 − 1 vertices and size |E(Γ(R))| = 1/2(3p^4 − 4p^3 − p^2 + 2). Using an equitable 4-cell partition induced by the ideal filtration ⟨ u^3 ⟩ ⊂ ⟨ u^2 ⟩ ⊂ ⟨ u ⟩, we prove that Spec_A(Γ(R)) decomposes into discrete levels λ∈ {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(λ) = 0, yielding adjacency energy ε(Γ(R)) = 4p^2 + 𝒪(p). In contrast, the Laplacian spectrum is entirely integral: Spec_L(Γ(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(Γ(R)) = ln p − 1/p + (ln p)/p + 𝒪((ln p)/p^2), with small-failure unreliability bounded by p^2(p − 1)(1 − q)^p−1. In linear consensus ẋ = −Lx, error trajectories decouple into three modal timescales { (p^3 − 1)^−1, (p^2 − 1)^−1, (p − 1)^−1 } with algebraic connectivity α(Γ(R)) = p − 1 and steady-state H_2 error variance H_2^2 = Kf(Γ(R))/2n^2 = 1/2p + 𝒪(p^−3). Furthermore, the Kirchhoff index yields a vanishing average resistance distance r̅(Γ(R)) = 2/p + 𝒪(p^−3) → 0, while unbiased Markovian random walks induce an asymptotic traffic concentration vector Π= (1/3, 0, 1/3, 1/3) with quadratic per-node core routing stress π_core/π_periph ∼ p^2 + p + 1.
ER  -