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We present bounds on two combinatorial properties of Cayley graphs in terms relating to the structure of their underlying group, Included in this work is a presentation of lower bounds on the diameter of Cayley graphs of groups with nilpotent subgrouPs and upper bounds on the size of node bisectors of Cayley graphs of groups with solvable subgroups.Cayley graphs, being endowed with algebraic structure, have been increasingly recognized as a source of interconnection networks underlying parallel computers. Their structure has been shown to endow parallel architectures with advantages, for example, in terms of algorithmic efficiency. Our results demonstrate limits on the communication power of certain classes of wellstructured interconnection networks.* A portion of this research was supported by National Science Foundation Grant CCR-88-12567 while both authors were attending the University of Massachusetts. 272F. Annexstein and M. Baumslag work of Margulis [24]) explicitly constructed expanders--an infinite family of graphs with the expansion property--by using two-dimensional linear functions. Klawe [20] added perspective to these results (of [16]) by showing that it was not possible to produce expanders using a restricted set of one-dimensional linear functions. Recently Lubotsky et al. [23] proved that a family of Cayley graphs--of complex algebraic structure--are "good ''1 expanders.Our motivation for this paper relates to a view of a number of authors, including [2], [3], and [10]- [14], who have argued for the use of Cayley graphs as the underlying interconnection network of parallel architectures. This is due to the fact that the symmetry of such graphs can often be usefully exploited in terms of algorithmic efficiency and fault tolerance. Many interconnection networks of theoretical and commercial importance are Cayley graphs of well-structured groups, including the torus, the supertoroidal, the butterfly, the cube-connectedcycles, and the hypercube. The communication capabilities of these networks are theoretically limited, since none of the underlying graphs possess the expansion property. We believe this work provides insight into understanding the tradeoffs between underlying group structure and connectivity properties of Cayley graphs. Our ResultsThe organization of this paper and an informal statement of our results is as follows. In Section 2 we describe the formal setting by providing the necessary graph-theoretic and group-theoretic notions. We use this formal framework to expose the role group structure plays in determining connectivity properties of Cayley graphs. This is accomplished by using structural properties of groups to determine the "rate of expansion" of associated graphs. We organize our work into two sets of results.The first set of results (Section 3) concerns the size of node bisectors. Theorem 1 gives an upper bound on the size of node bisectors of Cayley graphs of abelian groups. We extend this result to arbitrary Cayley graphs by analyzing the derived series of the underlying g...