  We study several problems related to the $\ell^p$ boundedness of Riesz
transforms for graphs endowed with so-called bounded Laplacians. Introducing a
proper notion of gradient of functions on edges, we prove for $p\in(1,2]$ an
$\ell^p$ estimate for the gradient of the continuous time heat semigroup, an
$\ell^p$ interpolation inequality as well as the $\ell^p$ boundedness of the
modified Littlewood-Paley-Stein functions for all graphs with bounded
Laplacians. This yields an analogue to Dungey's results in [Dungey08] while
removing some additional assumptions. Coming back to the classical notion of
gradient, we give a counterexample to the interpolation inequality hence to the
boundedness of Riesz transforms for bounded Laplacians for $1<p<2$. Finally, we
prove the boundedness of the Riesz transform for $1< p<\infty$ under the
assumption of positive spectral gap.
