Topology in Molecular Biology

12 The Spectral Geometry of Riemann Surfaces 211

Fig. 12.2.The graphX^2 ,^3 according to Brooks and Zuk

The simplest of them is the graphX^23 (Fig. 12.1). Here is a picture ofX^2 ,^3
taken from the book of Sarnak:
I think that you will agree that it is hard to say anything intelligent about
this graph from the picture. Indeed, Sarnak seems to have overlooked the fact
that the graph as he drew it was not even three-regular. We have supplied the
edge that Sarnak missed by writing it as a dotted line. While playing around
with some of the ideas which will follow, Andrzej Zuk and I noticed that one
can rearrange the graph in a way that is easier to understand. It is shown
below (Fig. 12.2).
In fact, all of the Ramanujan graphs have a nice structure that makes them
come out nice – not, perhaps, so nice as in this example, but still one that
suggests a reasonable geometric picture. The question is: what, if anything, is
this picture trying to tell us? Can we make good use of this extra structure?
Does it suggest a more general picture where there is more to the geometry of
graphs than meets the eye? While the investigations below came from other
sources, I think that a good way to understand where we are going is by
considering these questions in light of the example of the two ways of writing
the same graph.

12.3 The Noncompact Case

In this section, we want to extend our theorem on the behavior ofλ 1 under
coverings to the case where the base manifoldSis a Riemann surface of finite
area: