  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.
