The set of connective constants of Cayley graphs contains a Cantor space
英文摘要:
The purpose of this note is to prove that the set of connective constants of Cayley graphs contains a Cantor space.
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Sébastien Martineau
The purpose of this note is to prove that the set of connective constants of Cayley graphs contains a Cantor space.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
继续探索
For a transitive infinite connected graph $G$, let $μ(G)$ be its connective constant. Denote by $\mathbf{\cal G}$ the set of Cayley graphs for finitely generated infinite groups with an infinite-order generator which is independent of other generators. Assume $G\in\mathbf{\cal G}$ is a Cayley graph of a finitely presented group, and Cayley graph sequence $\{G_n\}_{n=1}^{\infty}\subset \mathbf{\cal G}$ converges locally to $G.$ Then $μ(G_n)$ converges to $μ(G)$ as $n\rightarrow\infty.$ This confirms partially a conjecture raised by Benjamini [2013. {\it Coarse geometry and randomness.} Lect. No…
The connective constant $μ(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the theorem was that the graphs support so-called 'unimodular graph height functions'. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of unimodular graph …
The connective constant of a graph is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. Strict inequalities are proved for connective constants of vertex-transitive graphs. Firstly, the connective constant decreases strictly when the graph is replaced by a non-trivial quotient graph. Secondly, the connective constant increases strictly when a quasi-transitive family of new edges is added. These results have the following implications for Cayley graphs. The connective constant of a Cayley graph decreases strictly when a new relator is added to the grou…
The connective constant $μ(G)$ of a quasi-transitive graph $G$ is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on $G$ from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph $G$. $\bullet$ We present upper and lower bounds for $μ$ in terms of the vertex-degree and girth of a transitive graph. $\bullet$ We discuss the question of whether $μ\geϕ$ for transitive cubic graphs (where $ϕ$ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vert…
The connective constant $μ(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. The relationship between connective constants and amenability is explored in the current work. Various properties of connective constants depend on the existence of so-called 'graph height functions', namely: (i) whether $μ(G)$ is a local function on certain graphs derived from $G$, (ii) the equality of $μ(G)$ and the asymptotic growth rate of bridges, and (iii) whether there exists a terminating algorithm for approximating $μ(G)$ to a give…
The connective constant $μ(G)$ of a graph $G$ is the asymptotic growth rate of the number of self-avoiding walks on $G$ from a given starting vertex. We survey three aspects of the dependence of the connective constant on the underlying graph $G$. Firstly, when $G$ is cubic, we study the effect on $μ(G)$ of the Fisher transformation (that is, the replacement of vertices by triangles). Secondly, we discuss upper and lower bounds for $μ(G)$ when $G$ is regular. Thirdly, we present strict inequalities for the connective constants $μ(G)$ of vertex-transitive graphs $G$, as $G$ varies. As a consequ…
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 investigate the validity of the inequality $μ\ge ϕ$ for infinite, transitive, simple, cubic graphs, where $ϕ:= \frac12(1+\sqrt 5)$ is the golden mean. The inequality is proved for several families of graphs including (i) Cayley graphs of infinite groups with three generators and strictly positive first Betti number, (ii) infinite, transitive, topologically locally finite (TLF) planar, cubic graphs, and (iii) cubic Cayley graphs with two ends. Bounds f…
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