# Is this lower bound for the size of minimal vertex cover new/interesting?

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!

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 $$\tau\left(G\right)\geq\frac{nd}{d-\lambda_{min}}$$