Neutrino mass matrices with one texture equality and one vanishing neutrino mass

In the light of latest data of neutrino oscillation experiments, we carry out a systematic investigation on the texture structures of Majorana neutrino mass matrix $M_{\nu}$, which contain one vanishing neutrino mass and an equality between two matrix elements. Among 15 logically possible patterns, it is found that for norm order ($m_{3}>m_{2}>m_{1}=0$) of neutrino masses only five of them are compatible with recent experimental data at the $3\sigma$ level, while for inverted order ($m_{2}>m_{1}>m_{3}=0$) ten patterns is phenomenologically allowed. In the numerical analysis, we perform a scan over the parameter space of all viable patterns to get a large sample of scattering points. We present the implications of each allowed pattern for three mixing angles $(\theta_{12},\theta_{2},\theta_{3})$, leptonic CP violation and neutrinoless double-beta decay, predicting strong correlations between oscillation parameters. The theoretical realization of a concrete example is discussed in the framework of Froggatt-Nielsen mechanism.