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The difference between the smallest eigenvalue and the next-smallest of a graph Laplacian (equivalently, the difference between the largest and next-largest of the random walk Markov chain on the graph) is the spectral gap, related to the Cheeger constant, etc. Is there any interpretation of the gap between the two extreme eigenvalues on the other side -- i.e. the largest ones of the Laplacian?

I am equally interested in the interpretation of the gap between the smallest two singular values of an arbitrary two matrix.

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    $\begingroup$ Hmm -- I must have a different convention than you when it comes to signs, because for me the spectral gap is the gap between the smallest two eigenvalues of a graph Laplacian (zero and the smallest positive one, if the graph is connected), not the two largest ones. $\endgroup$
    – Federico Poloni
    Commented Oct 8, 2017 at 15:32
  • $\begingroup$ For markov transition matrices, spectral gap is between 1 (largest e. value) and the next one (<1) $\endgroup$
    – Piyush Grover
    Commented Oct 8, 2017 at 17:48
  • $\begingroup$ I believe analogy with classical laplacian (or more correctly with Fokker-planck operators with ergodicity ) with neumann BC should serve well. The largest eigenvalues correspond to the fastest decaying eigenvectors...of course everything decays to the invariant distribution eventually. $\endgroup$
    – mystupid_acct
    Commented Oct 10, 2017 at 17:45

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