  In this note we study Landis conjecture for positive Schr\"odinger operators
on graphs. More precisely, we give a decay criterion that ensures when $
\mathcal{H} $-harmonic functions for a positive Schr\"odinger operator $
\mathcal{H} $ with potentials bounded from above by $ 1 $ are trivial. We then
specifically look at the special cases of $ \mathbb{Z}^{d} $ and regular trees
for which we get explicit decay criterion. Moreover, we consider the fractional
analogue of Landis conjecture on $ \mathbb{Z}^{d} $. Our approach relies on the
discrete version of Liouville comparison principle which is also proved in this
article.
