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Self-avoiding walks and connective constants. (arXiv:1704.05884v1 [math.CO])
来源于:arXiv
The connective constant $\mu(G)$ 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 $\mu$ in terms of the
vertex-degree and girth of a transitive graph.
$\bullet$ We discuss the question of whether $\mu\ge\phi$ for transitive
cubic graphs (where $\phi$ denotes the golden mean), and we introduce the
Fisher transformation for SAWs (that is, the replacement of vertices by
triangles).
$\bullet$ We present strict inequalities for the connective constants
$\mu(G)$ of transitive graphs $G$, as $G$ varies.
$\bullet$ As a consequence of the last, the connective constant of a Cayley
graph of a finitely generated group decreases strictly when a new relator is
added, and increases strictly when a non-trivial group element is declared to
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