AI 中文总结
研究了具有拉马努金界限的符号循环图，证明了特定签名的谱半径低于凯斯滕界，并探讨了四边形系统与磁通量的关系。
AI 中文摘要
对于具有n≥10个顶点且n为偶数的循环图C_n(1,2)，要求每个四边形都是不平衡的F_2系统是一致的，其解恰好形成四个切换类。我们显示包含在步-1边上为+1而在步-2边上为(-1)^i的签名的类具有谱{±2√(cos²θ_k + cos²2θ_k)}，其谱半径恰好为2√2，低于凯斯滕界2√3；四边形系统等价于交替三角形磁通量，因此四个类由(τ_0,α)坐标化，其谱半径仅取决于哈密顿环的holonomy α；并且两个扭曲类达到ρ_(n)=2√(cos²(π/n)+cos²(2π/n))<2√2。对n∈{8,10,12,14,16,18}的所有2^{n+1}个切换类进行穷举枚举显示ρ_(n)在每种情况下都是全局最小值，我们推测对于所有偶数n也是如此；下界是一个磁通量最小化陈述，意义在于利布的磁通-相位定理。对于奇数n，四边形系统是不一致的。
英文摘要
For the circulant graph $C_n(1,2)$ with $n\ge10$ even, the $\F_2$ system requiring every quadrilateral to be unbalanced is consistent and its solutions form exactly four switching classes. We show that the class containing the signing which is $+1$ on step-$1$ edges and $(-1)^i$ on step-$2$ edges has spectrum $\{\pm2\sqrt{\cos^2θ_k+\cos^2 2θ_k}\}$ and spectral radius exactly $2\sqrt2$, well below the Kesten bound $2\sqrt3$; that the quadrilateral system is equivalent to alternating triangle fluxes, so that the four classes are coordinatized by $(τ_0,α)$ and the spectral radius depends only on the Hamilton-cycle holonomy $α$; and that the two twisted classes attain $ρ_-(n)=2\sqrt{\cos^2(π/n)+\cos^2(2π/n)}<2\sqrt2$. Exhaustive enumeration of all $2^{n+1}$ switching classes for $n\in\{8,10,12,14,16,18\}$ shows that $ρ_-(n)$ is the global minimum in every case, and we conjecture this for all even $n$; the lower bound is a flux-minimization statement in the sense of Lieb's flux-phase theorem. For odd $n$ the quadrilateral system is inconsistent.
CommentsI combined this paper with another one (arXiv:2607.17343)
