Symmetric texture-zero mass matrices with eigenvalues quark mass

Working within the context of texture-zeros mechanism for fermionic mass matrices, we provide necessary and sufficient conditions on the characteristic polynomial coefficients such that it has real, simple and positive roots. We translate these conditions in terms of invariants from congruent matrices. Then, all symmetric texture-zero matrices are counted and classified. Next we apply the result from the first part to analyze the three, two and one zero texture matrices in a systematic way. Finally we solve analytically the $V_{ckm}$ mixing matrix for the four zero sets; we also analyze the $V_{ckm}$ for a particular case of four zero, four zero-perturbed and three zero sets.