  We show the integrated density of states for a variety of Jacobi operators on
graphs, such as the Anderson model and random hopping models on graphs with
Gaussian heat kernel bounds, can be estimated from above and below in terms of
the localization landscape counting function. Specific examples of these graphs
include stacked and decorated lattices, graphs corresponding to band matrices,
and aperiodic tiling graphs. The upper bound part of the landscape law also
applies to the fractal Sierpinski gasket graph. As a consequence of the
landscape law, we obtain landscape-based proofs of the Lifshitz tails in
several models including random band matrix models, certain bond percolation
Hamiltonians on $\mathbb{Z}^d$, and Jacobi operators on certain stacks of
graphs. We also present intriguing numerical simulations exploring the behavior
of the landscape counting function across various models.
