REU 2001
The students were: Sylvia Carlisle (Carlton College),
Brian Carty (Amherst College) Bejamin Chan (Rochester), Sawson
Eskander (Mount Holyoke College), Matt Noonan (Hampshire College),
Anne Short (Harvey Mudd College). Five of these students were
supported by NSF funds and one by Mount Holyoke funds. The
research this summer examined families of Ramanujan graphs, a class of optimally efficient graphs for the
transmission of information. These graphs are characterized by the
spectra associated to their adjacency matrices, and the only known
constructions are limited to k-regular graphs for $k = p + 1$, $p^n + 1$,
where p is a prime. Our problem this summer was to generalize these
constructions to k= pq and, especially, to k = 2p, that is, to see if
we might find a way of constructing expander graphs by gluing
together two underlying graphs known to be Ramanujan. The numerical
evidence suggests that these particular methods do yield expanders,
but not necessarily Ramanujan graphs. To work on the project,
students were required to learn about algebraic graph theory, to read
and understand the results for known k, to study Eichler's work on
general quaternion algebras and their maximal orders and Sarnak's book
on applications of modular forms. In addition, they learned how to
use new software for computational algebra and resolved various
complexity issues associated to the numerical problem of calculating
eigenvalues of graphs with many vertices. Finally, the group was able
to extend Chiu's work to the three additional definite quaternion
algebras with class number 1 that split at the prime p = 2.
The group's work was accepted for the undergraduate poster
session at the joint meetings in San Diego in January, 2002, but
because of scheduling conflicts, no member was available to attend.
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