The natural Next-to-Minimal Supersymmetric Standard Model (nNMSSM) is featured by predicting one $CP$-even Higgs boson satisfying $m_{h_1}\lesssim 120\mathrm{GeV}$ and Higgsinos lighter than about 300 GeV, and consequently, the cross section for dark matter (DM)-nucleon scattering in this scenario is usually quite large. We study the diphoton signal of the light Higgs boson in nNMSSM by considering the tight constraints from the latest LUX and PandaX-II experiments, and we conclude that the optimal value of the signal rate at 8 TeV LHC is greatly reduced in comparison with earlier predictions. For example, previous studies indicated that the rate may exceed 120 fb for $m_{h_1}\simeq 80\mathrm{GeV}$, while it is at most 25 fb if the lightest neutralino in the scenario is fully responsible for the measured DM relic density. We also investigate the case of $m_{h_1}\simeq 98\mathrm{GeV}$, which is hinted by the excesses of the large electron proton collider analysis on $Z\overline{b}b$ signal and the compact muon solenoid analysis on the diphoton signal. We conclude that nNMSSM can simultaneously explain the excesses at the $1\sigma$ level without violating any known constraints.

We systematically calculate various flavor-changing neutral-current top-quark processes induced by supersymmetry at the Large Hadron Collider, which include five decay modes and six production channels. To reveal the characteristics of these processes, we first compare the dependence of the rates for these channels on the relevant supersymmetric parameters, then we scan the whole parameter space to find their maximal rates, including all the direct and indirect current experimental constraints on the scharm-stop flavor mixings. We find that, under all these constraints, only a few channels, through $cg\to t$ at parton level and $t\to ch$, may be observable at the Large Hadron Collider.