Abstract
The connective constant of a transitive graph is the exponential growth rate of its number of self-avoiding walks. We prove that the set of connective constants of the so-called Cayley graphs contains a Cantor set. In particular, this set has the cardinality of the continuum.
Highlights
The connective constant of a transitive graph is the exponential growth rate of its number of self-avoiding walks
We prove that the set of connective constants of the so-called Cayley graphs contains a Cantor set
By Fekete’s Subadditive Lemma, the sequence c1n/n converges to some real number μ(G). This number does not depend on the choice of o and is called the connective constant of G
Summary
The connective constant of a transitive graph is the exponential growth rate of its number of self-avoiding walks. We prove that the set of connective constants of the so-called Cayley graphs contains a Cantor set. Given a group G and a finite generating subset S of G, the Cayley graph associated with (G, S) is the graph with vertex-set G and such that two distinct elements g and h of G are connected by an edge if and only if g−1h ∈ S ∪ S−1. This theorem implies the following result of Leader and Markström: the set of isomorphism classes of Cayley graphs has cardinality 2א0 .
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