The goal of this work is explore Weyl's Law as it may relate to the graphs underlying the Wolfram Model of Physics. Weyl's law describes the nature of the asymptotic growth of the eigenvalues of the Laplacian operator on bounded domains having Dirichlet and Neumann boundary conditions. The Laplace-Beltrami operator is a generalization of the Laplacian for operating on Riemannian manifolds. In the context of the Wolfram Model of physics, an ongoing area of investigation is whether the principles of Einstein's theory of General Relativity thereby arise as emergent phenomena. As General Relativity is described in the language of Riemannian Geometry, the question of whether properties of Riemannian manifolds also hold on these graphs. Weyl's law is a statement regarding the asymptotic growth of the eigenvalues of the Laplacian on bounded domains with DIrichlet or Neumann boundary conditions. The law has been extended to non-Euclidean space through a generalization of the Laplace operator called the Laplace-Beltrami operator. This investigation of Weyl's law in the context of the Wolfram Model posits a number of fundamental questions at the intersection of Riemannian geometry and graph theory, among which a principal question is what the appropriate analogue for the Laplace-Beltrami operator is for digraphs. In particular, does there exist a generalization of the graph Laplacian that encodes geometric information in the Wolfram model graphs analogous to Laplace-Beltrami operator on Riemannian manifolds.
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