Upper bound on the difference between two elements of an eigenvector Let $W$ be the non-negative, symmetric adjacency/affinity matrix for some connected graph. If $W_{ij}$ is large, then vertex $i$ and vertex $j$ have a heavily weighted edge between them. If $W_{ij} = 0$, then no edge connects vertex $i$ to vertex $j$. 
Now $L = \mathrm{diag}(W\mathbf{1})-W$ is the (unnormalized) graph Laplacian. Let $v$ be the Fiedler vector of $L$, that is, a unit eigenvector corresponding to the second smallest eigenvalue of $L$. As $W_{ij}$ increases, all else equal, $|v_i - v_j|$ tends to decrease---at least this is the idea behind spectral clustering. What is an upper bound on $|v_i - v_j|$, given quantities that don't require computing $v$, like $W_{ij}$ and $\|W\|$? 
Any suggestions or thoughts would be greatly appreciated.

Let me at least point out a trivial bound.
If $L$ is a Laplacian, then $x^TLx=\sum_{i,j}L_{i,j}(x_i-x_j)^2$. Let 
$\lambda$ be the eigenvalue for the Fiedler vector. Then 
$$\lambda = x^TLx = \sum_{i,j} w_{i,j}(x_i-x_j)^2,$$ whence 
$|x_i-x_j| \le (\lambda/w_{i,j})^{1/2}$.
