For a simple graph $G$ of order $n$, we define its adjacency matrix and Laplacian matrix. Both have real eigenvalues. Let $\theta_1(G) \geq \cdots \geq \theta_n(G)$ be the eigenvalues of the adjacency matrix and $\lambda_1(G) \geq \cdots \geq \lambda_n(G) = 0$ the eigenvalues of the Laplacian matrix of graph $G$. We study Nordhaus-Gaddum type inequalities for the eigenvalues of these two matrices. These are upper and lower bounds for sums of the forms $\theta_i(G) + \theta_i(\overline{G})$ and $\lambda_j(G) + \lambda_j(\overline{G})$, where $\overline{G}$ denotes the graph complement of $G$. The focus of this work is on the sums for the smallest eigenvalue of the adjacency matrix and the largest two eigenvalues of the Laplacian matrix.
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