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

Abstract

For an odd prime $p$, let $Γ(R)$ 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 $|E(Γ(R))| = \frac{1}{2}(3p^4 - 4p^3 - p^2 + 2)$. 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 $\operatorname{Spec}_A(Γ(R))$ decomposes into discrete levels $λ\in \{0, -1\}$ with multiplicities $p^3 - p^2 - 1$ and $p^2 - 3$, respectively, alongside three irrational roots of an irreducible cubic polynomial $P_3(λ) = 0$, yielding adjacency energy $ε(Γ(R)) = 4p^2 + \mathcal{O}(p)$. In contrast, the Laplacian spectrum is entirely integral: $\operatorname{Spec}_L(Γ(R)) = \{0^1, (p - 1)^{p^3 - p^2}, (p^2 - 1)^{p^2 - p - 1}, (p^3 - 1)^{p - 1}\}$. Consequently, the tree entropy satisfies $z(Γ(R)) = \ln p - \frac{1}{p} + \frac{\ln p}{p} + \mathcal{O}(\frac{\ln p}{p^2})$, 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 $α(Γ(R)) = p - 1$ and steady-state $H_2$ error variance $H_2^2 = \frac{Kf(Γ(R))}{2n^2} = \frac{1}{2p} + \mathcal{O}(p^{-3})$. Furthermore, the Kirchhoff index yields a vanishing average resistance distance $\overline{r}(Γ(R)) = \frac{2}{p} + \mathcal{O}(p^{-3}) \to 0$, while unbiased Markovian random walks induce an asymptotic traffic concentration vector $Π= (\frac{1}{3}, 0, \frac{1}{3}, \frac{1}{3})$ with quadratic per-node core routing stress $π_{\mathrm{core}}/π_{\mathrm{periph}} \sim p^2 + p + 1$.