Abstract
The connective constant μ of a graph G is the asymptotic growth rate of the number of self-avoiding walks on G from a given starting vertex. Bounds are proved for the connective constant of an infinite, connected, Δ-regular graph G. The main result is that \(\mu \geqslant \sqrt {\Delta - 1}\) if G is vertex-transitive and simple. This inequality is proved subject to weaker conditions under which it is sharp.
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Grimmett, G.R., Li, Z. Bounds on connective constants of regular graphs. Combinatorica 35, 279–294 (2015). https://doi.org/10.1007/s00493-014-3044-0
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DOI: https://doi.org/10.1007/s00493-014-3044-0