University of Vermont

cems
College of
Engineering and Mathematical Sciences

Department of Mathematics & Statistics

Research - Combinatorics and Graph Theory

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    Dan Archdeacon, Professor, works in graph theory--the study of networks. He specializes in embedding graphs on surfaces. This helps in understanding the underlying structure in graphs, and what makes them easy to draw. In turn, that helps in understanding the difficulty (or easiness) in laying-out computer networks. He is a leading researcher in topological graph theory and is managing editor of the highly respected Journal of Graph Theory.

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    Jeffrey Dinitz, Professor, works in the area of combinatorial designs. He is the managing editor-in-chief of the Journal of Combinatorial Designs as well as the co-editor of the CRC Handbook of Combinatorial Designs. Together with a number of different coauthors he has published numerous articles in many areas of combinatorial designs, including one-factorizations, Room Squares, triple systems, Latin squares, MOLS and transversal designs, graph decompositions, Howell designs, starters, balanced tournament designs, graph labeling, list coloring, and hill-climbing and orderly algorithms. He has also written papers on applications of combinatorial designs to networking and storage disk arrays. He has been on the faculty since 1980 and holds a secondary appointment in the Department of Computer Science.

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    Greg Warrington, Associate Professor, works in the area of algebraic combinatorics. It is often helpful to investigate families of algebraic and geometric objects by considering associated polynomials that encode some of their important structure. A non-mathematical example: One could encode the 13 children, 47 grandchildren, 94 great grandchildren, ... of Leonhard Euler by the polynomial
    1 + 13x + 47x2 + 94x3 + .... This throws away a lot of information (for example, names) but would be useful in budgeting for birthday presents. Greg is interested in two specific families of polynomials of this type that stem from questions in representation theory. In particular, he would like to find concrete combinatorial objects whose variations are counted by the coefficients of these polynomials. Finding such objects can uncover new relationships and insights that, while beautiful in their own right, can also help explain the underlying algebra and geometry.