On the limitations of the use of solvable groups in Cayley graph cage constructions

Authors:
Marston Conder;Geoffrey Exoo;Robert Jajcay
Affiliations:
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand;Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN 47809, USA;Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN 47809, USA
Venue:
European Journal of Combinatorics
Year:
2010

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Abstract

A (k,g)-cage is a (connected) k-regular graph of girth g having smallest possible order. While many of the best known constructions of small k-regular graphs of girth g are known to be Cayley graphs, there appears to be no general theory of the relationship between the girth of a Cayley graph and the structure of the underlying group. We attempt to fill this gap by focusing on the girth of Cayley graphs of nilpotent and solvable groups, and present a series of results supporting the intuitive notion that the closer a group is to being abelian, the less suitable it is for constructing Cayley graphs of large girth. Specifically, we establish the existence of upper bounds on the girth of Cayley graphs with respect to the nilpotency class and/or the derived length of the underlying group, when this group is nilpotent or solvable, respectively.