REU 2007

Central Michigan University

Final Reports

Here are final reports for the four undergraduate research projects developed during the NSF supported Research Experience for Undergraduates at Central Michigan University, June 4, 2006 – July 28, 2006.  Some projects developed here were also supported by an NSF grant for “long-term” undergraduate research.

 

·        On the minimum semidefinite rank of a graph using vertex sums, graphs with msr(G) = |G|-2 and the msrs of certain graph classes (pdf)

Jonathan Beagley, Eileen Radzwion, Sara Rimer, Rachael Tomasino, Jennifer Wolfe, Andrew Zimmer, under the direction of Sivaram Narayan 

Given a Hermitian matrix A, associate a simple, undirected graph G(A) where V(G) is labeled by the rows and columns of A and the edge set corresponds to the nonzero entries of A. The collection of all Hermitian matrices that share a common graph G is denoted H(G). We consider subset P(G) of all positive semidefinite matrices corresponding to the graph G. The minimum semidefinite rank of G denoted msr(G), is defined to be the minimum rank among all matrices in P(G).  In this paper we present results on finding the msr(G) when G is written as a vertex sum of two graphs G1 and G2 that share a cut set of at most two vertices.  A classification of all graphs with  msr(G)=|G|-2  is given.  An upper bound for  msr(G) when G is obtained from G1 and G2 by cancellation of edges in a star forest is also presented.  Moreover, results on the msr of cartesian products, strong products, corona, and line graphs of certain graphs are proven.

·        Evaluating El-Niño's effect in the mid-Michigan area and Puerto Rico using multiple linear regression models (pdf)  (Figures)

Emily Baker, Maida Cabezudo, Kelan Hlavaty, Dana Kell and Sarah Nowak, under the direction of John Daniels 

We use data collections of sea surface temperatures in the El Niño 3.4 region off the West Coast of South America to evaluate air temperatures in the Mid-Michigan area and Puerto Rico.  We present two competing linear models to explain the air temperatures in Mid-Michigan based on sea surface temperature and one model that analyzes El Niño's effect in Puerto Rico.

·        Distance Regular Cayley Graphs (pdf)

Fatema Burhani and Adam Giambrone, under the direction of Ken Smith

Which distance regular graphs are also Cayley graphs?  If a graph is a Cayley graph Cay(S;G) in an abelian group G, then the eigenvalues of the graph (and their multiplicities) correspond to the character values (and their multiplicities) of the dual group G* acting on the set S.  An idempotent construction in the group ring is then used to examine possible parameters of distance regular Cayley graphs in abelian groups.

·        Determining Existence of Hadamard Difference Sets in Groups of Order 144 (pdf) (LaTex)

Nicole Kroeger, Marcus Miller, Christopher Mooney and Kathleen Shepard, under the direction of Ken Smith

We present the results of an investigation into (144,66,30) difference sets and also a powerful new construction that allows us to classify 37 unresolved cases. This spread construction is, roughly, a generalization of the more common product constructions. Out of 197 groups of order 144, we were able to determine existence in 170 of groups, finding difference sets in 131 and ruling out difference sets in 39. If we assume a conjecture by Eric Lander stating that a group with a cyclic Sylow-3 subgroup has no difference set, then only 1 group of order 144 remains.