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I have found this lower bound for the size of minimal vertex cover (and proved it).

If a simple connected graph G on n vertices has largest and smallest eigenvalues $\lambda_1,\lambda_n$, respectively, and $\theta_{n-1}$ is the second smallest Laplace eigenvalue, then $$ \tau(G)\geq\frac{n\theta_{n-1}^{2}}{\theta_{n-1}^{2}-4\lambda_{1}\lambda_{n}} $$ When $\tau(G)$ is the minimal size of minimal vertex cover.

I checked it for some graphs and it was very tight for most of them. There are some graphs for which the estimation is not close, but I think it works pretty well most times.

I tried to look for it and didn't find anything. Anyone saw that before? Are there better bounds?

Thank!

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I found a better bound for regular graphs- the Hoffmann bound:

Let G be a d-regular graph on $n$ vertices with minimal eigenvalue $\lambda_{min}$. Then $$ \alpha\left(G\right)\leq\frac{-n\lambda_{min}}{d-\lambda_{min}} $$

It is known that for a graph $G$ on $n$ vertices, $\alpha\left(G\right)+\tau\left(G\right)=n$. Thus $$ \tau\left(G\right)\geq\frac{nd}{d-\lambda_{min}} $$

For regular graphs, this bound is always tighter than mine.

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