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For the first time, a calibration between the He  i |$\lambda 5876$| /H  |$\beta$| emission line ratio and the helium abundance y = 12 + log(He/H) for Narrow line regions of Seyfert 2 Active Galactic Nuclei is proposed. In this context, observational data (taken from the SDSS-DR15 and from the literature) and direct abundance estimates (via the |$T_{\rm e}$| -method) for a sample of 65 local (⁠|$z \: \lt \: 0.2$|⁠) Seyfert 2 nuclei are considered. The resulting calibration estimates the y abundance with an average uncertainty of 0.02 dex. Applying our calibration to spectroscopic data containing only strong emission lines, it yields a helium abundance distribution similar to that obtained via the |$T_{\rm e}$| -method. Some cautions must be considered to apply our calibration for Seyfert 2 nuclei with high values of electron temperature (⁠|${\gtrsim} 20\, 000$|  K) or ionization parameter (⁠|$\log U \ \gt\ {-}2.0$|⁠). [ABSTRACT FROM AUTHOR]

We analyse the validity of optical diagnostic diagrams relying on emission-lines ratios and in the context of classifying Active Galactic Nuclei (AGNs) according to the cosmic metallicity evolution in the redshift range |$0 \le z \le 11.2$|⁠. In this regard, we fit the results of chemical evolution models (CEMs) to the radial gradients of the N/O abundances ratio derived through direct estimates of electron temperatures (T e-method) in a sample of four local spiral galaxies. This approach allows us to select representative CEMs and extrapolate the radial gradients to the nuclear regions of the galaxies in our sample, inferring in this way the central N/O and O/H abundances. The nuclear abundance predictions for theoretical galaxies from the selected CEMs, at distinct evolutionary stages, are used as input parameters in AGN photoionization models built with the Cloudy code. We found that standard BPT diagnostic diagrams are able to classify AGNs with oxygen abundances |$\rm 12+\log (O/H) \gtrsim 8.0$| [ |$(Z/Z_{\odot }) \gtrsim 0.2$| ] at redshift |$z \lesssim 4$|⁠. On the other hand, the He  ii λ4685/Hβ versus [N  ii ]λ6584/Hα diagram produces a reliable AGN classification independent of the evolutionary stage of these objects. [ABSTRACT FROM AUTHOR]