For construction of a sequence of discrete Laplacians whose spectrum converges to that of a given Riemannian manifold, see
Dodziuk, Jozef, Finite-difference approach to the Hodge theory of harmonic forms, Am. J. Math. 98, 79-104 (1976). ZBL0324.58001.
A good survey on spectral graph theory is
Colin de Verdière, Yves, Spectres de graphes, Cours Spécialisés. 4. Paris: Société Mathématique de France. vi, 114 p. (1998). ZBL0913.05071.
(There are surely more recent surveys or lecture notes, maybe somebody else can point at them.)
Eigenvalues of the graph Laplacian carry interesting combinatorial information, such as the expansion parameter, see Wikipedia articles Expander graphs and Kirchhoff's theorem. In order to approximate the geometry of a manifold, it is appropriate to use weights on the edges. The so-called cotangent Laplacian (well-known in dimension 2 and having an analog in higher dimensions) is a good choice.