发表机构
Department of Computer Science. Indiana University, Bloomington（印第安纳大学布卢明顿分校计算机科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究广义煎饼图 $P(m,n)$ 的归一化拉普拉斯谱间隙，确定其在 $n\to\infty$ 和 $m\to\infty$ 时的渐近阶，推翻了 Blanco 和 Buehrle 关于其构成扩张器族的猜想。
AI 中文摘要
广义煎饼图 $P(m,n)$ 是由广义前缀翻转生成的彩色置换群 $\mathbb{Z}_m\wr S_n=(\mathbb{Z}_m)^n\rtimes S_n$ 的凯莱图。本文证明，对所有 $m,n\geq2$，其归一化拉普拉斯的谱间隙 $\gamma(P(m,n))$ 满足 $\alpha_m/n\leq\gamma(P(m,n))\leq1/n$，其中 $\alpha_m$ 是仅依赖于 $m$ 的正常数。由此可得，对每个固定的 $m\geq2$，当 $n\to\infty$ 时，$\gamma(P(m,n))$ 为 $\Theta_m(1/n)$。证明结合了 Cesi 的半递归谱间隙不等式与颜色-位置陪集 Schreier 图相关算子的傅里叶分解。对固定的 $n\geq2$，还证明当 $m\to\infty$ 时，$\gamma(P(m,n))$ 为 $\Theta_n(m^{-2})$，这推翻了 Blanco 和 Buehrle 关于固定 $n$ 时对应无向广义煎饼图构成扩张器族的猜想。
英文摘要
The generalized pancake graph $P(m,n)$ is the Cayley graph of the group of colored permutations $\mathbb{Z}_m\wr S_n=(\mathbb{Z}_m)^n\rtimes S_n$ generated by generalized prefix reversals. In this paper, we establish that, for all $m,n\geq2$, the spectral gap $γ(P(m,n))$ of the normalized Laplacian satisfies $α_m/n\leqγ(P(m,n))\leq1/n$, where $α_m$ is a positive constant that depends only on $m$. As a consequence, for every fixed $m\geq2$, $γ(P(m,n))$ is $Θ_m(1/n)$ as $n\to\infty$. The proof combines Cesi's semi-recursive spectral-gap inequality with a Fourier decomposition of the appropriate operators associated with a coset Schreier graph of color-position pairs. For fixed $n\geq2$, we also establish that $γ(P(m,n))$ is $Θ_n(m^{-2})$ as $m\to\infty$. This disproves a conjecture of Blanco and Buehrle asserting that, for fixed $n$, the corresponding undirected generalized pancake graphs form an expander family. Additionally, we present a counterexample to a recent conjecture of Greaves and Zhu concerning equality between the spectral gaps of the full Cayley graph and the associated coset Schreier graph.
CommentsFixed typos, included a new conjecture
