As they say in the abstract, this retroactively explains the diversity of discrete Laplacians in the literature. I don't know to what extent this result applies to Chung and Yau's setup.

The paper "Discrete Laplace operators: no free lunch" by Wardetzky, Mathur, K{\"a}lberer, and Grinspun (cs.columbia.edu/cg/pdfs/1180993110-laplacian.pdf) discusses several nice properties of the Laplacian (symmetry, positive semi-definiteness, maximum principle, kernel containing the constant functions, and two others) and shows that no linear operator on functions defined on a triangular mesh can have all of these properties. (continued)