Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4595294 | Journal of Number Theory | 2008 | 13 Pages |
Abstract
Let a and b be positive and relatively prime integers. We show that the following are equivalent: (i) d is a dead end in the (symmetric) Cayley graph of Z with respect to a and b, (ii) d is a Frobenius value with respect to a and b (it cannot be written as a non-negative or non-positive integer linear combination of a and b), and d is maximal (in the Cayley graph) with respect to this property. In addition, for given integers a and b, we explicitly describe all such elements in Z. Finally, we show that Z has only finitely many dead ends with respect to any finite symmetric generating set. In Appendix A we show that every finitely generated group has a generating set with respect to which dead ends exist.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory