发表机构
Cambridge University（剑桥大学）
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Grigorchuk群的Cayley图的连接常数，证明其下界大于黄金分割数φ，扩展了关于无限顶点传递立方图连接常数的猜想，并分析了轨道Schreier图的标签序列出现比例。
AI 中文摘要
图G的连接常数μ(G)是从给定顶点出发的自回避行走数的指数增长率。我们对由序列ω∈{0,1,2}^N编码的一般Grigorchuk群的Cayley图G_ω的连接常数证明了上下界。特别地，对于任意此类Cayley图（满足ω的简单条件），有μ(G_ω) > φ，其中φ:=1/2(1+√5)为黄金分割数。这扩展了作者与钟阳Li在《立方图与黄金分割》（Discrete Math. 343 (2020), 文章111638）中的早期工作，该工作曾猜想所有无限、顶点传递、立方图都满足μ(G)≥φ。当前工作还包括对一般Grigorchuk群的轨道Schreier图中给定标签序列出现比例的分析。
英文摘要
The connective constant $μ(G)$ of a graph $G$ is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We prove upper and lower bounds for the connective constants of Cayley graphs $G_ω$ of a general Grigorchuk group encoded by a sequence $ω\in\{0,1,2\}^{\Bbb N}$. In particular, $μ(G_ω) > ϕ$ for any such Cayley graph (subject to a simple condition on $ω$), where $ϕ:= \frac12(1+\sqrt 5)$ is the golden mean. This extends earlier work of the author and Zhongyang Li in "Cubic graphs and the golden mean'', Discrete Math. 343 (2020), article 111638, where it was conjectured that $μ(G)\geϕ$ for all infinite, vertex-transitive, cubic graphs. The current work includes an analysis of the proportions of appearances of given label-sequences in the orbital Schreier graphs of general Grigorchuk groups.
Commentsv2: includes improved numerical bounds
