The relation between the diagonal entries and the eigenvalues of a symmetric matrix, based upon the sign pattern of its off-diagonal entries

It is known that majorization is a complete description of the relationships between the eigenvalues and diagonal entries of real symmetric matrices. However, for large subclasses of such matrices, the diagonal entries impose much greater restrictions on the eigenvalues. Motivated by previous results about Laplacian eigenvalues, we study here the additional restrictions that come from the off-diagonal sign-pattern classes of real symmetric matrices. Each class imposes additional restrictions. Several results are given for the all nonpositive and all nonnegative classes and for the third class that appears when n = 4. Complete description of the possible relationships are given in low dimensions.