NSF-REU 2005 at Central Michigan

REU 2005

Central Michigan University

Final Reports

Here are seven final reports on undergraduate research projects from the NSF supported Research Experience for Undergraduates at Central Michigan University, June 6, 2005 – July 30, 2005.

The Minimum PSD Rank of a Graph (pdf)

Yunjiang Jiang, Kseniya Kudryavtseva, and Janeta Marinova, under the direction of Sivaram Narayan.

An investigation into the minimum rank of positive semi-definite Hermitian matrices of finite graphs.

An Upper Bound to the Pebbling Number of a Graph (pdf)

Joyce Charlesworth, Theang Ho, Brian Keinath and David Lin, under the direction of Sivaram Narayan.

A result on the pebbling number for graphs of radius r is generalized to a large class of graphs of diameter d. In addition results on (r,m) pebbling and the 2-pebbling property are presented.

Zero Divisor Semigroup Graphs (pdf)

Yunjiang Jiang, Cleland Loszewski and Erica Purdy, under the direction of Lisa DeMeyer.

Finite commutative semigroups with zero divisors are studied via the complement of the zero divisor graph. Results are given on semigroups whose zero divisor graph is a triangulation of a compact surface.

Difference Sets of Order 25 (pdf)

Strom Borman under the direction of Ken Smith.

Several open parameters for nonabelian difference sets with k-lambda = 25 are resolved in the negative, using algebraic number theory and irreducible group representations of degrees two or four.

Walk Regular Graphs Which Are Not Vertex Transitive (pdf)

Graph Voyager -- A computerized exploration of graphs (pdf)

Katherine Soller under the direction of Ken Smith; Alan LaMielle under the direction of Paul Albee and Ken Smith;

Katherine Soller and Alan LaMielle develop a project in algebraic graph theory. Katherine Soller studies walk regular graphs, graphs with the property that the number of n-cycles on a vertex depends only on n and is independent of the vertex. Vertex transitive graphs are walk regular. Walk regular graphs which are not vertex transitive are investigated, focusing on graphs with four or five eigenvalues.

Alan LaMielle creates a Java application to investigate the eigenvalues and other attributes of finite graphs. The program allows the user to create or modify graphs and compute a variety of graph properties in real time, as the user changes the graph. This tool was used to support some of the other investigations listed above.

Cellular Automata (pdf)

Dustin Gage, Elizabeth Laub, Briana McGarry, under the direction of Ken Smith.

Wolfram’s Rule 30 Cellular Automata is analyzed using a variety of statistical tests. Rule 30 passes most randomness tests but the results vary with window size. Global properties were also examined and show trends that suggest even window sizes have weaknesses and should be avoided until more information is known. Other cellular automata similar to Rule 30 are also examined. The various tests and CA rules are summarized in an Excel spreadsheet.