# Normalized Laplacian of graph with empty rows

Let $G = (V, W)$ be a graph, $V$ is the set of vertices and $W$ is the adjacency matrix. I want to compute the normalized Laplacian $L_n = D^{-1/2} L D^{-1/2}$, where L is the combinatorial Laplacian $L = D - W$ and $D$ is the row sum diagonal matrix defined by $$d_{ij} = \begin{cases} \sum_{j=1}^{n} w_{ij}, &i = j,\\ 0,&i \neq j. \end{cases}$$ The problem I have is that my adjacency matrix has some empty rows, which I cannot remove, since the corresponding columns are not empty, but if I allow $d_{ii}$ to be 0 for some $i$, then I cannot build $L_n$ anymore because $0^{-1/2}$ is not defined. Is there a standard trick to overcome this issue?

• Is the graph unoriented? If so the fact that you have an empty row means that the graph is not connected, and I think that in that case you should work on each connected component separately. Oct 16, 2016 at 13:49
• The graph is oriented, otherwise I could remove the row and the corresponding column. There is no point in defining the Laplacian in this setting, maybe? Oct 16, 2016 at 14:32
• See my answer below for more on this. Why are you using a graph laplacian here? Perhaps there's something else you can use to solve your problem. Oct 18, 2016 at 22:31