The graph Laplacian is ussually defined as $L=D-A$, where $D$ is the degree matrix and $A$ the adjacency matrix. It gets its name from being the discrete analog of the Laplacian operator from calculus.
One can clearly see the common points discretizing the continuous Laplacian. In 1D it is discretized as the standard second derivative, as follows: $$\nabla^2 f = f_{i+1} + f_{i-1} -2f_{i}.$$
Analogously, in 2D: $$\nabla^2 f = f_{i+1,j} + f_{i-1,j} + f_{i,j+1} + f_{i,j-1} -4f_{i,j}.$$
Note that a grid of dimension $d$ can be regarded as a regular graph, where each node has a degree $d$ (i.e., is connected to $d$ neighbors). In the above expressions, the first terms in the r.h.s. are of the form $Af$, while the last term is $Df=df$. Thus, one can generalize the above expressions to arbitrary grids of dimension $d$, as: $$\nabla^2 f = A f - D f = (A-D) f.$$
This is the same than the standard graph Laplacian $L=D-A$ with the exception of the sign. My question is: why the sign in the standard definition of the graph Laplacian is the opposite with respect to the continuous analogy? Is it just a convention, or is there any mistake in the analogy outlined above?