Let  denote the set of real symmetric matrices whose entries are in the interval . For an real symmetric matrix we always denote the eigenvalues of in decreasing order by () (). We will study the smallest eigenvalue () and the largest eigenvalue () when varies in . Constantine  proved that if , then 2i f is even, 12i f is odd. So the matrices treated there are nonnegative. The proof techniques of  are graph- theoretic. In  Roth gave another proof of this result by analysis of eigenvectors.