We present the results of a dust-reverberation survey of quasars at redshifts z<0.6. We found a delayed response of the K-band flux variation after the optical flux variation in 25 out of 31 targets, and obtained the lag time between them for 22 targets. Combined with the results for nearby Seyfert galaxies, we provide the largest homogeneous collection of K-band dust-reverberation data for 36 type 1 active galactic nuclei (AGNs). This doubles the sample and includes the most distant AGN and the largest lag so far measured. We estimated the optical luminosity of the AGN component of each target using three different methods: spectral decomposition, the flux-variation-gradient method, and image decomposition. We found a strong correlation between the reverberation radius for the innermost dust torus and the optical luminosity over a range of approximately four orders of magnitude in luminosity, as is already known for Seyfert galaxies. We estimated the luminosity distances of the AGNs based on their dust-reverberation lags, and found that the data in the redshift-distance diagram are consistent with the current standard estimates of the cosmological parameters. We also present the radius-luminosity relations for isotropic luminosity indicators such as the hard X-ray (14--195 keV), [OIV] 25.89 um, and mid-infrared (12 um) continuum luminosities, which are applicable to obscured AGNs.
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