Cactus Trees, and Estimations of the Spectral Radius of VertexTransitive Graphs
Abstract
This paper gives lower bounds on the spectral radius of vertextransitive 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 coefficientwise 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 1skeletons of tessellations of hyperbolic plane by 4ggons, 4g per vertex). We have for example for g=2 0.662420<=M<=0.662816, and for g=3 0.552773<=M<=0.552792.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 2001
 arXiv:
 arXiv:math/0112108
 Bibcode:
 2001math.....12108B
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Combinatorics;
 20F65;
 20F69;
 05C38;
 20F04
 EPrint:
 12 pages, 1 figure