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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 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. 

In this work, we study the Anderson model on the Sierpinski gasket graph. We first identify the almost sure spectrum of the Anderson model when the support of the random potential has no gaps. We then prove the existence of the integrated density states of the Anderson model and show that it has Lifshitz tails with a Lifshitz exponent determined by the ratio of the volume growth rate and the random walk dimension of the Sierpinski gasket graph.