Cubic graphs and the golden mean

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 := 1 2 (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.

Self-avoiding walk on nonunimodular transitive graphs

“…Recently, the study of SAW on more general graphs has gathered momentum. In particular, a systematic study of SAW on transitive graphs has been initiated in a series of papers by Grimmett and Li [11,15,19,17,16,13,14], which is primarily concerned with properties of the connective constant. Other works on SAW on non-Euclidean transitive graphs include [12,33,3,29,32,10].…”

Presentations for Vertex Transitive Graphs

“…In this section we construct an example of a 2-ended cubic vertex transitive graph which is not a Cayley graph. This answers a question of Watkins [Wat90] also appearing in [GL20].…”

“…Incidentally, we find a cubic (i.e. 3-regular) 2-ended vertex transitive graph which is not a Cayley graph, answering a question of Watkins [Wat90], recently revived by Grimmett & Li [GL20]. Although this construction does not explicitly use the theory developed in this paper, our study of partite presentations helped us understand where to look for such examples.…”

Site Percolation on Planar Graphs

“…The characterization of one-ended vertex-transitive planar graphs as tilings of natural geometries makes it possible to develop universal techniques to study statistical mechanical models on all these graphs; see [11], for example, about a universal lower bound of connective constants on all the infinite, connected, transitive, planar, cubic graphs. 2.2.…”