Given a graph, one can write down the adjacency matrix for the graph; its first eigenvalue must be positive; scale the matrix so that the first eigenvalue is one. Now there is a theorem, known as the spectral gap theorem (there are parallel theorems that I'm not totally familiar with) which says that the difference between the first and second eigenvalue must be at least some number (on the order of 5% if I recall; I don't have a good reference handy).
I went to a colloquium where someone was collecting data which could be made to essentially look like a graph; they would they test for the dimensionality of the data by looking at the eigenvalues of this matrix and seeing when the eigenvalues dropped off such that the variance was very low. however, depending on the distribution of eigenvalues the cutoff point may be arbitrary. At the time, she said that a spectral gap for later eigenvalues would be useful, for making cutoff points less arbitrary (i.e. having a way to know if the next eigenvalue is definitively NOT a repeated eigenvalue because it's too far).
This isn't exactly my specialty so I'm sorry if my explanation is a little rough.
Ok, I've never used such an approach; I don't think I've ever worked with any data that could reasonably be made to look like a graph. (Unless perhaps it was raw detector hits before being reconstructed into tracks; and I've only brushed the edge of that sort of thing.) As for dimensionality, I would usually just count the variables. We are clearly talking about something very different from what I usually do.
In response to falenas108's "Ask an X" thread. I have a PhD in experimental particle physics; I'm currently working as a postdoc at the University of Cincinnati. Ask me anything, as the saying goes.
This is an experiment. There's nothing I like better than talking about what I do; but I usually find that even quite well-informed people don't know enough to ask questions sufficiently specific that I can answer any better than the next guy. What goes through most people's heads when they hear "particle physics" is, judging by experience, string theory. Well, I dunno nuffin' about string theory - at least not any more than the average layman who has read Brian Greene's book. (Admittedly, neither do string theorists.) I'm equally ignorant about quantum gravity, dark energy, quantum computing, and the Higgs boson - in other words, the big theory stuff that shows up in popular-science articles. For that sort of thing you want a theorist, and not just any theorist at that, but one who works specifically on that problem. On the other hand I'm reasonably well informed about production, decay, and mixing of the charm quark and charmed mesons, but who has heard of that? (Well, now you have.) I know a little about CP violation, a bit about detectors, something about reconstructing and simulating events, a fair amount about how we extract signal from background, and quite a lot about fitting distributions in multiple dimensions.