The Discrete Mathematics Seminar at Iowa State University is an eclectic mix of topics, including graph theory, combinatorics, linear algebra and abstract algebra. Presentations vary with the speaker and include the speaker's research, related research by others, and expository talks.  Many of the expository talks (typically labelled as "Introduction to") are suitable for interested graduate students and faculty who are not specialists in the area.

• Fall 06
  
• Spring 06
• Fall 05

Y. T. Poon, Feb. 27, Product of differences of complex numbers and the determinantal conjecture.
Let R = {a_1, ..., a_n}, S = {c_1, ..., c_n} be two sets of complex numbers, and
$$T = \left\{ \prod\limits_{j=1}^n \(a_j -c_{\sigma(j)}\): \sigma\in S_n\right\},$$
where S_n denotes the set of all permutations of {1,2,...,n}. We will give  some necessary and
sufficient conditions for  T to be a subset of a straight line in the complex
plane. The set T is related to the Marcus and de Oliveira conjecture on determinant.

Chi-Kwong Li, March 6, Spectrum preserving maps
We discuss recent results on spectrum preserving maps on
operators.  Previous research on this topic often impose assumptions on the maps such as linearity, additivity, multiplicatively, etc.  We show that most of these assumptions can be removed, and one can  still obtain nice structure for spectrum preserving maps.

Fan Chung, March 27, Random graphs and Internet graphs
We will discuss some recent developments on random graphs with given expected degree distributions.Such ramdom graphs can be used to model various very large graphs arising in Internet and telecommunications. In turn, these "massive graphs" shed insights and lead to new directions for random graph theory. For example, it can be shown that the sizes of connected components depend primarily on the average degree and the second-order average degree under certain mild conditions. Furthermore, the spectra of the adjacency matrices of some random power law gra phs obey the power law while the spectra of the Laplacian follow the semi-circle law. We will mention a number of related results and problems that a re suggested by various applications of massive graphs.