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We have investigated the low-frequency 1/f noise of both suspended andon-substrate graphene field-effect transistors and its dependence on gatevoltage, in the temperature range between 300K and 30K. We have found that thenoise amplitude away from the Dirac point can be described by a generalizedHooge's relation in which the Hooge parameter {\alpha}H is not constant butdecreases monotonically with the device's mobility, with a universal dependencethat is sample and temperature independent. The value of {\alpha}H is alsoaffected by the dynamics of disorder, which is not reflected in the DCtransport characteristics and varies with sample and temperature. We attributethe diverse behavior of gate voltage dependence of the noise amplitude to therelative contributions from various scattering mechanisms, and to potentialfluctuations near the Dirac point caused by charge carrier inhomogeneity. Thehigher carrier mobility of suspended graphene devices accounts for values of1/f noise significantly lower than those observed in on-substrate graphenedevices and most traditional electronic materials.

Mobility-dependent Low-frequency Noise in Graphene Field Effect Transistors