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## Estimating a spectral gap

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?

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Do you have any other information about A? As stated, the difference that you want to bound can be made arbitrarily small. – Vidit Nanda Oct 22 at 11:19
Well, $A$ is in fact a signless Laplacian of a graph. – Felix Goldberg Oct 22 at 11:37
Searching google for "arbitrarily small spectral gap" produces this paper: <www.mis.mpg.de/fileadmin/jjost/abj21-6-04.pdf> which claims to construct graphs of arbitrarily small spectral gap given mild conditions on the degrees of the vertices. What is your starting point? If you know each entry of $A$ then you can presumably compute the eigenvalues and hence the gap. On the other hand if you want a theorem that bounds the gap for some general $A$, then there is no hope without more assumptions on the graph. – Vidit Nanda Oct 22 at 12:43
One could get bounds depending on some feature of $A$. For instance, we all know that $a^2+b^2$ can get arbitrarily small for positive $a$ and $b$, but nevertheless it is interesting to discover the bound $2ab\leq a^2+b^2$. – Federico Poloni Oct 22 at 13:50