REU 2006

Central Michigan University

Final Reports

Here are final reports for the five undergraduate research projects developed during the NSF supported Research Experience for Undergraduates at Central Michigan University, June 5, 2006 – July 29, 2006.

Distance-Regular Cayley Graphs (pdf)

Appendix (Excel file)

Bernadette Boyle and Raena Bryant, under the direction of Ken Smith.

A survey of distance-regular Cayley graphs, focusing on graphs of diameter 3.

Non-existence of some cyclic difference sets (pdf)

Bridget Franklin and Steven Sam, under the direction of Ken Smith

Applying rational idempotents of the group ring and techniques from algebraic number theory, we resolve the existence/non-existence of cyclic difference sets in six previously open parameter sets. [June 2007. We have discovered a gap in one of the proofs; we will publish a correction at a later date.]

On Polya's Orchard Problem (pdf)

Alexandru Hening and Michael Kelly, under the direction of Boris Bekker

We examine Polya’s orchard problem for two-dimensional compact convex orchards. We obtain results for rhombus-shaped orchards in two dimensions. We also develop estimates for the minimal tree radius in the natural generalization to three dimensions.

Linear Algebra of Magic Squares (pdf)

Michael Lee, Elizabeth Love and Elizabeth Wascher, under the direction of Sivaram Narayan

The vector space dimension of regular and pandiagonal magic squares are computed. We produce both singular and nonsingular matrices for odd order regular magic squares and provide a simple proof that all even order regular magic squares are singular.

On the minimum vector rank of a multigraph (pdf)

Ian Rogers, under the direction of Sivaram Narayan

We present methods for determining the minimum vector rank (msr) of a multigraph and provide general lower and upper bounds for mvr(G). We relate the minimum vector rank to the minimum semidefinite rank of a graph.