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In my understanding spectral theory including the study of the
eigenvalues of the Laplacian operator does apply to arbitrary graphs
(and not only to lattices). In fact, this is done in the context of
the algebraic graph theory (cf., Godsil & Royle as well as the older
Cvetkovic, Doob & Sachs). Now from the other point of mathematical
view you've mentioned, due to discreteness, there is no underlying
continuity to enable any proper differential geometry of graphs. The
only approach that I've seen to be somehow close is the procedure of
gluing together a number of fiber bundles (as circle bundles in the
case of Waldhausen's graph manifolds or in Seifert fiber spaces) in
order to employ combinatorics in the geometrical classification of
manifolds. But, I guess, this approach is exactly the other way around
of what you're looking for.
Let me also add - just in case you're not already familiar with this -
on spectral theory in social networks you may want to read the work of
the late Bill Richards and Andrew Seary:
On Thu, Jul 16, 2009 at 5:27 AM, Rojas, Fabio
Guillermo<[log in to unmask]> wrote:
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> I am not sure if this is the right list, but here's my question and it comes from studying social science network data:
> I have a graph (G) and a real valued function (F) on the vertices (V). Is there a way to study the "surface" defined by (v, f(v)) in GxR? In other words, how do I translate the ides of multivariable calculus and differential geometry into the discrete world?
> A google search does show that that there is a literature (e.g., there is the discrete Laplacian), but it mainly applies to mesh grids (ie, you chop up the domain of the function into a grid), but not for an arbitrary graph, which is my issue. Any tips? What would I need to know about calculus on graphs that might be different than the continuous version?
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