The interlacing theorem tells us that ***for rank 1 $W$*** the eigenvalues of $M \pm W$ are sandwiched between those of $M$ (see Corollary 4.3.9 [here][1]). Therefore 
$null(M)-1 \leq nullity(M \pm W) \leq null(M)+1$ and this is equivalent to $r(M)-1 \leq r(M \pm W) \leq r(M)+1$.

These bounds are tight. For the case of graphs there are some detailed studies of when the different cases obtain.


  [1]: https://books.google.co.il/books?id=O7sgAwAAQBAJ&pg=PA243&lpg=PA243&dq=matrix%20analysis%20rank%20one%20interlacing&source=bl&ots=lUMbzifvi3&sig=jiUurTyRbBNn22_OtXniwllgnEk&hl=en&sa=X&ei=YmrQVOzQB4KuUcvIgMgI&ved=0CCcQ6AEwAQ#v=onepage&q=matrix%20analysis%20rank%20one%20interlacing&f=false