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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?

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