rate of blood flow and size of blood vessels?

For a given pressure difference along the length of the blood vessel, or any tube, the flow will be greater if the diameter is greater.  In fact, the flow is proportional to the fourth power of the radius.

Rate and velocity are similar, but not the same (rate is quite a vague concept):

The rate of blood flow (e.g. in ml/sec) through a larger vessel (given the same mean arterial pressure) would be greater, but the velocity (i.e. the change in distance of a mass of blood cells in a given time) would be less.

For example, Transcranial Doppler studies (TCDs) use ultrasound to measure the flow velocity in cerebral blood vessels ... the *higher* the flow velocity, the *narrower* the artery.  (TCDs are used to detect cerebral vasospasm).

TCDs are so sensitive because, as scottsdalehigh64 correctly stated, flow is inversely proportional to the 4th power of the radius [of the vessel].