Cactus Trees, and Estimations of the Spectral Radius of Vertex-Transitive Graphs

Abstract

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 and underestimation of its spectral radius. The bounds obtained are very good for the Cayley graph of surface groups of genus g>=2, with standard generators (these graphs are the 1-skeletons of tessellations of hyperbolic plane by 4g-gons, 4g per vertex). We have for example for g=2 0.662420<=|M|<=0.662816, and for g=3 0.552773<=|M|<=0.552792.