what is the significance of the inverse of an adjacency matrix?

Question

Suppose I have a graph and I calculate the eigenvalues of the adjacency matrix and find that there are some number of zero eigenvalues. Do zero eigenvalues have any significance? Also is there a good way to interpret the inverse (when it exists)? In some cases the inverse is called the Green's function. Thanks!

Answer 1

So I'm not sure if this the sort of answer you want, but I think it's kind of interesting. (Fair warning: I haven't really thought about this much before either, mostly just spitballing. Hopefully it's not total nonsense.)

Remember that powers of an adjacency matrix count the number of walks of a certain size from one vertex to another. Therefore if we interpret the graph as a network, and initialize with a vector of "starting" data, the adjacency matrix will tell us how that data propagates through the network.

A zero eigenvalue shows that the nullspace is nontrivial, and so there is a nonzero starting vector which is sent to zero: we can interpret this as a "sink": in this scenario it is as if the users all jacked out with their data and left none to continue flowing. Moreover, with general initialization, any arrangement of data that looks like this will continue to be removed at every step.

We might naively expect that a network with a zero-value eigenvector will show a strong dispreference for data arrangements of this type, but this doesn't seem quite right: the arrangement will generally appear and disappear constantly, since being in the kernel of course does not prevent you from being in the image. Moreover, dispreference is not unique to zero-value eigenvalues, but any $\varepsilon$-value eigenvectors are disprefered as well, at least relative to the dominant trend corresponding to the largest eigenvalues.

So what it seems like this arrangement really is, is some sort of short term, private behavior. Trends in private content, like most emails, cease-and-desist letters, bomb threats, etc. would exist in the kernel, whereas trends in public content, like Youtube videos, free music, chain letters, etc. would exist outside of it.