Coauthors: AIM minimum rank-special graphs work group, Wayne Barrett, Mireille Bousquet-Melou, Joe Buhler, Fan Chung, Kevin Costello, Warwick de Launey, Erik Demaine, Ron Graham, Jason Grout, Gerhard Guettler, Mohammad Hajiaghayi, H. Tracy Hall, Paul Horn, Pavel Karasik, Robert Kleinberg, Miklos Laczkovich, Tom Leighton, Colin Mallows, Jia Mao, John Sinkovic, Wasin So, Colin Starr, Tomohiro Tachi, Eric Tressler, Amy Yielding, Nan Zang

Miscellaneous--research

Eigenvalues and structures of graphs -- Ph.D. dissertation written at UC San Diego under the direction of Fan Chung. Much of this has shown up in various papers I have published. Bounding the number of graphs containing very long induced paths -- Masters thesis written at Brigham Young University under the direction of Wayne Barrett.

JAVA applets written to look at what happens to daughters when bisecting triangles using some well defined central point. This is related to the papers Iterated triangle partitions and The lost daughters of Gergonne. The papers have generated many beautiful images and some can be found in the blog of Loic Seguin-Charbonneau "Triangulations creatives".

For the paper Finding patterns avoiding many monochromatic constellations we generated a Maple worksheet that gives a method to compute the coefficient of a block pattern given a constellation. We also have an extended Maple worksheet that contains more information about good block colorings. There is also an interactive JAVA applet that allows you to see what happens to the coefficient of a pattern as we perturb the pattern and/or the constellation. In addition we have the JAVA programs used to generate block patterns for 4-APs as well as the general pattern [0,q,1].

For the paper Irreducible Apollonian configurations and packings we generated a Maple worksheet that automates finding the center and radii of a packing placed in standard position. We also have a list of all irreducible configuration of circles with between 10 and 15 circles that can be used in the worksheet. Finally, there is also a collection of packings with various curvatures marked in the circles.

While visiting The Center for Combinatorics at Nankai University in September 2006 I gave three lectures that served as an introduction to the field of graph theory. The first lecture was an introduction to three common spectra that looked at the adjacency matrix, the Laplacian matrix, and the normalized Laplacian and why each one might be used. The second lecture was about applications of Courant-Fischer and gave some basic applications including examining what happens with the removal of a subgraph. The third lecture discussed Cheeger constants and discrepancy and looked at various properties of edge expansion in graphs.

I have posed various problems which have appeared. These include American Mathematical Monthly #11030, #11265 and Mathematics Magazine #1668, #1730, #1761, #Q1008.

In Fall of 2004 Fan Chung taught a course about the combinatorics of patterns in subsets and graphs. I worked to collect and collate the notes for the course into a PDF document; of course Fan did all the hard work of teaching the course.

A collection of the simplest aperiodic strongly connected directed graphs and their spectrums under the definition of the normalized Laplacian (as defined by Chung).

Miscellaneous unpublished notes: Relating the arboricity with the chromatic number of a graph Determining the underlying functions for Cauchy power and exponential forms A property of the gcd; and The Moebius transform of the triangular numbers The art of juggling with two balls A property of positive semidefinite matrices Tangent line transformations Method for doing the (approximate) Markov chains for bisection

A letter to the editor of my old school about the slogan do the math used for a fundraising campaign.

Talks/presentations