Is the laplacian operator for meshes just the sum of the differences of neighbouring vertices?
You can see why the sum of differences is giving a second derivative if you look at the derivative of a one-dimensional function $f$. The derivative of $f$ at $x_0$ is
$$ \frac{df}{dx}(x_0) \approx \frac{f(x_0 + dx) - f(x_0)}{dx} $$
The second derivative is then
\begin{eqnarray*}
\frac{d^2f}{dx^2}(x_0) &amp;\approx&amp; 
  \frac{df/dx(x_0) - df/dx(x_0-dx)}{dx} \\
  &amp;\approx&amp; \frac{\left(f(x_0+dx) - f(x_0)\right) -
                  \left(f(x_0) - f(x_0-dx)\right)}{dx^2} \\
  &amp;=&amp; \frac{\left(f(x_0+dx) - f(x_0)\right) +
                  \left(f(x_0-dx) - f(x_0)\right)}{dx^2},
\end{eqnarray*}
i.e. the sum of the local differences, scaled by a distance factor $dx^2$. In a graph Laplacian, you don't have any distances, just graph connections, so it's just the sum of the local differences.
When working on meshes, though, we typically want to take the geometry of the mesh into account. There are many ways to do this, but the most popular seems to be the cotangent Laplacian, where the local difference for each edge is weighted by the sum of the cotangents of the angles opposite that edge.