  The "slope-number" of a graph $G$ is the minimum number of distinct edge
slopes in a straight-line drawing of $G$ in the plane. We prove that for
$\Delta\geq5$ and all large $n$, there is a $\Delta$-regular $n$-vertex graph
with slope-number at least $n^{1-\frac{8+\epsilon}{\Delta+4}}$. This is the
best known lower bound on the slope-number of a graph with bounded degree. We
prove upper and lower bounds on the slope-number of complete bipartite graphs.
We prove a general upper bound on the slope-number of an arbitrary graph in
terms of its bandwidth. It follows that the slope-number of interval graphs,
cocomparability graphs, and AT-free graphs is at most a function of the maximum
degree. We prove that graphs of bounded degree and bounded treewidth have
slope-number at most $O(\log n)$. Finally we prove that every graph has a
drawing with one bend per edge, in which the number of slopes is at most one
more than the maximum degree. In a companion paper
(http://arxiv.org/abs/math/0606450), planar drawings of graphs with few slopes
are also considered.
