  We consider the localization landscape function $u$ and ground state
eigenvalue $\lambda$ for operators on graphs. We first show that the maximum of
the landscape function is comparable to the reciprocal of the ground state
eigenvalue if the operator satisfies certain semigroup kernel upper bounds.
This implies general upper and lower bounds on the landscape product
$\lambda\|u\|_\infty$ for several models, including the Anderson model and
random hopping (bond-disordered) models, on graphs that are roughly isometric
to $\mathbb{Z}^d$, as well as on some fractal-like graphs such as the
Sierpinski gasket graph. Next, we specialize to a random hopping model on
$\mathbb{Z}$, and show that as the size of the chain grows, the landscape
product $\lambda\|u\|_\infty$ approaches $\pi^2/8$ for Bernoulli off-diagonal
disorder, and has the same upper bound of $\pi^2/8$ for Uniform([0,1])
off-diagonal disorder. We also numerically study the random hopping model when
the band width (hopping distance) is greater than one, and provide strong
numerical evidence that a similar approximation holds for low-lying energies in
the spectrum.
