We study general implications of the IceCube observations in the energy range from $10^{6}$ GeV to $10^{10}$ GeV for the origin of extragalactic ultrahigh energy cosmic rays assuming that high energy neutrinos are generated by the photomeson production of protons in the extragalactic universe. The PeV-energy neutrino flux observed by IceCube gives strong bounds on the photomeson-production optical depth of protons in their sources and the intensity of the proton component of extragalactic cosmic rays. The neutrino flux implies that extragalactic cosmic-ray sources should have the optical depth greater than $sim 0.01$ and contribute to more than a few percent of the observed bulk of cosmic rays at 10 PeV. If the spectrum of cosmic rays from these extragalactic sources extends well beyond 1 EeV, the neutrino flux indicates that extragalactic cosmic rays are dominant in the observed total cosmic-ray flux at 1 EeV and above, favoring the dip transition model of cosmic rays. The cosmic-ray sources are also required to be efficient neutrino emitters with the optical depth close to unity in this case. The highest energy cosmic-ray ($sim 10^{11}$ GeV) sources should not be strongly evolved with redshift to account for the IceCube observations, suggesting that any cosmic-ray radiation scenarios involving distant powerful astronomical objects with strong cosmological evolution are strongly disfavored. These considerations conclude that none of the known extragalactic astronomical objects can be simultaneously a source of both PeV and trans-EeV energy cosmic rays. We also discuss a possible effect of cosmic-ray propagation in magnetized intergalactic space to the connection between the observed total cosmic-ray flux and neutrino flux.
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