Bounds on connective constants of regular graphs

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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Correspondence to Geoffrey R. Grimmett.

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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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Mathematics Subject Classification (2000)

  • 05C30
  • 82B20