  $\delta$-hyperbolic graphs, originally conceived by Gromov in 1987, occur
often in many network applications; for fixed $\delta$, such graphs are simply
called hyperbolic graphs and include non-trivial interesting classes of
"non-expander" graphs. The main motivation of this paper is to investigate the
effect of the hyperbolicity measure $\delta$ on expansion and cut-size bounds
on graphs (here $\delta$ need not be a constant), and the asymptotic ranges of
$\delta$ for which these results may provide improved approximation algorithms
for related combinatorial problems. To this effect, we provide constructive
bounds on node expansions for $\delta$-hyperbolic graphs as a function of
$\delta$, and show that many witnesses (subsets of nodes) for such expansions
can be computed efficiently even if the witnesses are required to be nested or
sufficiently distinct from each other. To the best of our knowledge, these are
the first such constructive bounds proven. We also show how to find a large
family of s-t cuts with relatively small number of cut-edges when s and t are
sufficiently far apart. We then provide algorithmic consequences of these
bounds and their related proof techniques for two problems for
$\delta$-hyperbolic graphs (where $\delta$ is a function $f$ of the number of
nodes, the exact nature of growth of $f$ being dependent on the particular
problem considered).
