A Note on the Growth Rate of Planar Graphs.
Neal Brand, Margaret J. Morton · 1996
If \\Gamma is a planar, locally finite, vertex transitive, 1-ended graph, then there is a particular `niceness' about the arrangement of the regions incident to a vertex in \\Gamma. Using this feature, it can be shown that \\Gamma can be embedded in either the Euclidean plane or the hyperbolic plane in such a way that every edge has the same length and every angle in an n-cycle bounding a region has the same measure. Moreover, there is a simple condition which tells whether \\Gamma is embedded in the Euclidean plane or the hyperbolic plane. The geometry of these two planes is then exploited to show that \\Gamma must have either quadratic growth or exponential growth depending on which plane it is embedded in. 1. Introduction. The graphs considered in this paper are simple, infinite, and planar. The symbols V (\\Gamma), E (\\Gamma) and aut(\\Gamma) will denote respectively, the vertex set, the edge set, and the automorphism group of \\Gamma. All graphs will be assumed to be locally finite; th...