  The sum of the absolute values of the eigenvalues of a graph is called the
energy of the graph. We study the problem of finding graphs with extremal
energy within specified classes of graphs. We develop tools for treating such
problems and obtain some partial results. Using calculus, we show that an
extremal graph ``should'' have a small number of distinct eigenvalues. However,
we also present data that shows in many cases that extremal graphs can have a
large number of distinct eigenvalues.
