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

A (k,g)(k,g)-cage is a (connected) kk-regular graph of girth gg having smallest possible order. While many of the best known constructions of small kk-regular graphs of girth gg 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.