Cactus trees and lower bounds on the spectral radius of vertex-transitive graphs

This paper gives lower bounds on the spectral radius of vertex-transitive graphs, based on the number of "prime cycles" at a vertex. The bounds are obtained by constructing circuits in the graph that resemble "cactus trees", and enumerating them. Counting these circuits gives a coefficient-wise underestimation of the Green function of the graph, and hence an underestimation of its spectral radius. The bounds obtained are very good for the Cayley graph of surface groups of genus g greater than or equal to 2 with standard generators (these graphs are the I-skeletons of tessellations of hyperbolic plane by 4g-gons, 4g per vertex). We have for example for g = 2 0.662418 less than or equal to parallel toMparallel to less than or equal to 0.662816, and for g = 3 0.552773 less than or equal to parallel toMparallel to less than or equal to 0.552792.