For an adjacency matrix $A$ that represent a graph $G=\langle V,E\rangle$, I need to show that the maxcut is bounded by:
$ \mathrm{maxcut} \leq \frac{1}{2}|E| - \frac{|V| \lambda_{\min}(A)}{4}, $
where $\lambda_{\min}(A)$ is the minimal eigenvalue of $A$.
I tried to use the fact that $\lambda_{\min}(A) = \min(\frac{X^TAX}{X^TX}) $
and to add that to the fact that $ X^TAX = 2 \sum_{ i,j \in E}^{a} \left( x_{i}x_{j} \right) $ for all vector $ x \in R^{|v|}$, but it doesn't work.
Any ideas?