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We present the first statistical survey of the properties of the 12CO(1-0) and 12CO(3-2) line emission from the nuclei of a nearly complete subsample of 60 infrared (IR) luminous galaxies selected from SCUBA Local Universe Galaxy Survey (SLUGS). This subsample is flux limited at S60 μm ≥ 5.24 Jy with far-IR (FIR) luminosities mostly at LFIR > 1010 L☉. We compare the emission line strengths of 12CO(1-0) and (3-2) transitions at a common resolution of ~15''. The measured 12CO(3-2) to (1-0) line intensity ratios r31 vary from 0.22 to 1.72, with a mean value of 0.66 for the sources observed, indicating a large spread of the degree of excitation of CO in the sample. These CO data, together with a wide range of data at different wavelengths obtained from the literature, allow us to study the relationship between the CO excitation conditions and the physical properties of gas/dust and star formation in the central regions of galaxies. Our analysis shows that there is a nonlinear relation between CO and FIR luminosities, such that their ratio LCO/LFIR decreases linearly with increasing LFIR. This behavior was found to be consistent with the Schmidt law relating star formation rate to molecular gas content, with an index N = 1.4 ± 0.3. We also find a possible dependence of the degree of CO gas excitation on the efficiency of star-forming activity. Using the large velocity gradient (LVG) approximation to model the observed data, we investigate the CO-to-H2 conversion factor X for the SLUGS sample. The results show that the mean value of X for the SLUGS sample is lower by a factor of 10 compared to the conventional value derived for the Galaxy, if we assume the abundance of CO relative to H2, ZCO = 10-4. For a subset of 12 galaxies with H I maps, we derive a mean total face-on surface density of H2 + H of about 42 M☉ pc-2 within about 2 kpc of the nucleus. This value is intermediate between that in galaxies like our own and those with strong star formation.