Abstract
The connective constant μ(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 μ≥ϕ for infinite, transitive, simple, cubic graphs, where ϕ≔12(1+5) is the golden mean. The inequality is proved for several families of graphs including (i) Cayley graphs of infinite groups with three generators and strictly positive first Betti number, (ii) infinite, transitive, topologically locally finite (TLF) planar, cubic graphs, and (iii) cubic Cayley graphs with two ends. Bounds for μ are presented for transitive cubic graphs with girth either 3 or 4, and for certain quasi-transitive cubic graphs.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
