Suppose you have a real positive definite matrix $A$ who eigenvalues are $\lambda_{1} \leq \lambda_{2} \leq \ldots \leq \lambda_{n}$. I am interested in bounding from below $\lambda_{2}-\lambda_{1}$. (It is known that $\lambda_{1}$ is simple). Are there any known methods for this kind of problem?
[Math] Estimating a spectral gap
matricesmatrix-theoryspectral-gap
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Best Answer
You mention that $A$ is the Laplacian of a graph. The field of spectral graph theory is well established, and I think there are several techniques that can be used to bound the spectral gap of a graph Laplacian for an undirected weighted graph.
Perhaps the most straightforward is known as Cheeger's inequality which controls the spectral gap, $\lambda_2 - \lambda_1$ by the Cheeger constant of the graph. See the wikipedia entry for Cheeger constant: http://en.wikipedia.org/wiki/Cheeger_constant_(graph_theory)
If you might also consider trying a Log sobolev inequality: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoap/1034968224
I hope this helps.