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Math Project Descriptions 2011

Algebraic Graph Theory Group  Dr. Sung-Yell Song
Existence and Nonexistence of (Directed) Strongly Regular Graphs

Strongly regular graphs are important objects in modern algebraic graph theory. The interest in strongly regular graphs has been stimulated by the development of the theory of finite permutation groups and the classification
of finite simple groups. The strongly regular graphs arise from various area of mathematics including group theory, association schemes, finite geometries, combinatorial designs and algebraic codes.

A strongly regular graph with parameters (v, k, λ, μ) is defined as an undirected regular graph G with v vertices satisfying the property: “the number of common neighbors of vertices x and y is k if x = y, λ if x and y are adjacent, and μ if x and y are non-adjacent vertices.” A loopless directed graph D with v vertices is called directed strongly regular graph with parameters (v, k, t, λ, μ) if and only if D satisfies the following conditions:
i) Every vertex has in-degree and out-degree k.
ii) Every vertex x has t out-neighbors that are also in-neighbors of x.
iii) The number of directed paths of length two from a vertex x to another vertex y is λ if there is an edge from x to y, and is μ if there is no edge from x to y.

There are many problems related to the characterization and classification of these graphs. We plan to settle some of the existence or non-existence problems for given parameter sets while we investigate various sources of (directed) strongly regular graphs. Sample problems that we might be tempted to explore include:
• How many strongly regular graphs with parameters (64, 28, 12, 12) are there?
• Is there a directed strongly regular graph with parameters (24, 10, 5, 3, 5)?
• Prove that there is no directed strongly regular graph with parameters (30, 7, 5, 0, 2).
• Characterize all tactical configurations that yield the directed strongly regular graphs with parameters ((n² − 1)(n³ − 1)/(n − 1)², n(n + 1), n, n − 1, n).


Dynamical Systems and Markov Chains  Dr. Wolfgang Kliemann

Many ‘real world’ phenomena can be modeled mathematically via ordinary differential equations; this includes many engineering systems, reaction kinetics in chemistry, population dynamics in biology, market behavior in economics, etc. More often than not, these systems will be subjected to perturbations (cars traveling on roads, variations of influx into a chemical reactor, environmental conditions, etc) that need to be taken into account to obtain a workable model of the phenomenon.  If we have statistical information about the perturbation (such as spectra of road surfaces, weather predictions, or market volatility), a finite state Markov chain in discrete time is often the model of choice for the perturbation.

We are then faced with a (continuous time) differential equation (on a continuous state space) that is perturbed by a (discrete time) Markov chain (with finitely many, hence discrete states). How can one build a mathematical theory that studies these (hybrid) systems? What can be said about the behavior of such systems, and about our ability to control them (i.e. to make the system behave like we want it to behave)? How can we estimate from actual data the parameters of Markov chains that model specific perturbations well?

The project will address at least some of these questions; its focus will depend on the interest and backgrounds of participants. We will start looking at mathematical descriptions of such hybrid systems (including their simpler version, switched systems, that omit the probabilistic part) and at numerical methods that allow reliable simulations of system behavior. We will then study either control theoretic aspects, or more statistical questions of parameter estimation.


Matrices over C[a, b] Dr. Justin Peters

We will begin by looking at the basic notions of Banach space and Banach algebra. We will examine what the spectral radius formula and spectral mapping theorem tell us in the case of the familiar examples of C[a, b], the continuous complex-valued functions on the interval [a, b], and M_n, the complex n × n matrices.

After reviewing central ideas from matrix theory (i.e., eigenvalues, eigenvectors, the Cayley-Hamilton theorem, diagonalization) we take up the major theme of this project: matrices over C[a, b], that is, matrices whose entries are in C[a, b]. This is a Banach algebra, which we denote M_n(C[a, b]), which combines the features of the two familiar examples in new and interesting ways. What are the appropriate notions of eigenvector and eigenvalue in this setting? How can the spectral radius be computed? How far can the analogy between M_n
and M_n(C[a, b]) be taken? For example, if A is an element of this Banach algebra such that for every t 2 [a, b] the matrix A(t) is diagonalizable by a unitary matrix in Mn, is A diagonalizable by a unitary
in M_n(C[a, b])?


Zero forcing/graph propagation  Dr. Leslie Hogben, Dr. Michael Young

Initially a subset Z of the vertices of a graph G are colored black and the remaining vertices are colored white.  The color change rule is that if a black vertex v has exactly one white neighbor w, then change the color of w to black.   The set Z is a zero forcing set if after applying the color change rule until no more changes are possible, all the vertices of G are black.  The zero forcing number is the minimum size of a zero forcing set.  Zero forcing arose in the study of determining the minimum rank/maximum nullity among real symmetric matrices having nonzero off-diagonal positions described by the edges of a given graph (the zero forcing number is an upper bound for maximum nullity, and independently in the study of control of quantum systems in physics, where it is called graph infection or propagation.   This project will investigate problems related to the zero forcing number.

Students involved in this project will be part the ISU Combinatorial Matrix Theory Research Group; more information is available on that page. This group regularly publishes its results.  The summer 2004, 2005, 2006, 2009, and 2010 groups all published papers (see list of papers).

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