发表机构
Department of Mathematics, Visva-Bharati; Indian Institute of Technology Madras（维斯瓦-巴拉蒂大学数学系; 印度马德拉斯理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文刻画了有限局部环上零因子图的谱性质，并分析其网络共识、可靠性与传输动力学，揭示代数连通度与路由应力等关键指标。
AI 中文摘要
对于奇素数 $p$，设 $\Gamma(R)$ 为与有限局部环 $R = \mathbb{F}_p[x]/\langle x^4 \rangle$ 相关联的零因子图，该图有 $p^3 - 1$ 个顶点，边数 $|E(\Gamma(R))| = \frac{1}{2}(3p^4 - 4p^3 - p^2 + 2)$。利用由理想滤过 $\langle u^3 \rangle \subset \langle u^2 \rangle \subset \langle u \rangle$ 诱导的等量 4 胞腔划分，我们证明 $\operatorname{Spec}_A(\Gamma(R))$ 分解为离散能级 $\lambda \in \{0, -1\}$，其重数分别为 $p^3 - p^2 - 1$ 和 $p^2 - 3$，此外还有不可约三次多项式 $P_3(\lambda) = 0$ 的三个无理根，从而得到邻接能量 $\epsilon(\Gamma(R)) = 4p^2 + \mathcal{O}(p)$。相比之下，拉普拉斯谱完全为整数：$\operatorname{Spec}_L(\Gamma(R)) = \{0^1, (p - 1)^{p^3 - p^2}, (p^2 - 1)^{p^2 - p - 1}, (p^3 - 1)^{p - 1}\}$。因此，树熵满足 $z(\Gamma(R)) = \ln p - \frac{1}{p} + \frac{\ln p}{p} + \mathcal{O}(\frac{\ln p}{p^2})$，小故障不可靠性以 $p^2(p - 1)(1 - q)^{p-1}$ 为界。在线性共识 $\dot{x} = -Lx$ 中，误差轨迹解耦为三个模态时间尺度 $\{ (p^3 - 1)^{-1}, (p^2 - 1)^{-1}, (p - 1)^{-1} \}$，代数连通度 $\alpha(\Gamma(R)) = p - 1$，稳态 $H_2$ 误差方差 $H_2^2 = \frac{Kf(\Gamma(R))}{2n^2} = \frac{1}{2p} + \mathcal{O}(p^{-3})$。此外，Kirchhoff 指标给出趋于零的平均电阻距离 $\overline{r}(\Gamma(R)) = \frac{2}{p} + \mathcal{O}(p^{-3}) \to 0$，而无偏马尔可夫随机游走诱导渐近流量集中向量 $\Pi = (\frac{1}{3}, 0, \frac{1}{3}, \frac{1}{3})$，每节点核心路由应力二次增长 $\pi_{\mathrm{core}}/\pi_{\mathrm{periph}} \sim p^2 + p + 1$。
英文摘要
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 $π_{\mathrm{core}}/π_{\mathrm{periph}} \sim p^2 + p + 1$.
