Published on *Computer Science Department at Princeton University* (http://www.cs.princeton.edu)

In this article we discuss several very simple graphs which have fairly small eigenvalues. Perhaps the most important result is that one of the graphs is a Cayley graph whose corresponding Cayley hypergraphs have a second eigenvalue which is essentially as small as a Cayley hypergraph can have. Also, these

graphs are very simple to write down, and the eigenvalue calculation is based on the trace method and well known results about exponential sums.