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!