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I'm looking for the spectral gap of common graphs (alternatively, the mixing time of a (lazy) random walk on these graphs). Asymptotic values are fine. Assume that every node has a sufficient number of self-loops. (For regular graphs at least d many) Spectral gaps should be for example,

  • Clique $O(1)$
  • Star $O(1)$
  • Expanders $O(1)$
  • Binary tree $\Theta(1/n)$
  • Cycle $\Theta(1/n^2)$
  • Grid $\Theta(1/n)$

What about other classes? For example d-nary trees or graphs arising from preferential attachment (I believe these should have $O(1)$). I'm especially interested in graphs where the gap between conductance and spectral gap is very high (Does this happen when the degree is small?).

Cheers

choco_addicted
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Xarph
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1 Answers1

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There is a nice table in Aldous & Fill page 169 including many quantities of interest. Most notably the relaxation time $\tau_2$, which is defined as the inverse of the spectral gap.

Xarph
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