Cubic graphs and the golden mean

Geoffrey R. Grimmett, Zhongyang Li

Published 2016-10-01Version 1

The connective constant $\mu(G)$ 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 $\mu \ge \phi$ for infinite, transitive, simple, cubic graphs, where $\phi:= \frac12(1+\sqrt 5)$ is the golden mean. The inequality is proved for several families of graphs including all infinite, transitive, topologically locally finite (TLF) planar, cubic graphs. Bounds for $\mu$ are presented for transitive cubic graphs with girth either $3$ or $4$, and for certain quasi-transitive cubic graphs.