We consider {\em monotone} embeddings of a finite metric space into low
dimensional normed space. That is, embeddings that respect the order among the
distances in the original space. Our main interest is in embeddings into
Euclidean spaces. We observe that any metric on n points can be embedded into
l2n, while, (in a sense to be made precise later), for almost every
n-point metric space, every monotone map must be into a space of dimension
Ω(n).
It becomes natural, then, to seek explicit constructions of metric spaces
that cannot be monotonically embedded into spaces of sublinear dimension. To
this end, we employ known results on {\em sphericity} of graphs, which suggest
one example of such a metric space - that defined by a complete bipartitegraph.
We prove that an δn-regular graph of order n, with bounded diameter
has sphericity Ω(n/(λ2+1)), where λ2 is the second
largest eigenvalue of the adjacency matrix of the graph, and 0 < \delta \leq
\half is constant. We also show that while random graphs have linear
sphericity, there are {\em quasi-random} graphs of logarithmic sphericity.
For the above bound to be linear, λ2 must be constant. We show that
if the second eigenvalue of an n/2-regular graph is bounded by a constant,
then the graph is close to being complete bipartite. Namely, its adjacency
matrix differs from that of a complete bipartite graph in only o(n2)
entries. Furthermore, for any 0 < \delta < \half, and λ2, there are
only finitely many δn-regular graphs with second eigenvalue at most
λ2