Yes they do. Wide tires which are used by dragsters come in different sturdiness or durometer ratings. If the tires are soft, there will be more initial friction. You will observe that the use of these wide tires has no importance with the friction.
When the car is in full stop, the tires’ diameter is small. Once the tires are rolling, they were made in such a way that the centripetal force allow the diameter of the tire to expand. This causes some favorable effects to shift the final drive gear ratio to higher speeds.
Racers are meticulous in determining the size of tires to have an optimum increase in diameter over a specific rotational acceleration of the wheel. This makes the race avail of the very limited narrow torque curve of their engines which are moving the gears frequently. This reason is more acceptable than the width/friction reason.
Friction is surface independent in some cases but not in all aspects especially for tires. Based on engineering, the friction force is equivalent to the force giving pressures to the 2 surfaces together. This may be applicable only to certain conditions and objects. The constant factor of proportion is called coefficient of friction.
The coefficient of friction depends on the component and condition of both surfaces having friction. If the surface becomes hard and adherent, that means that friction theory is not working. There must be large friction forces if there is lack of forces pushing the 2 surfaces together. When the surfaces adhere together when you force them to separate, a negative static friction coefficient may occur.
Therefore, the forces of friction of the tires are not related to the size.
Drag racers have 1 speed transmission that slips the clutch during the race. Dragsters prefer the right width, height and tire compound that provides the maximum friction during the race.
Therefore it is clear that the forces of friction of the tires are not related to the size. The reason why tires used in drag racing are wide is to make them firm not to slip when spinning and the car is heading to different directions. What measures when the tires tend to slip is the point when static friction is overcome by sliding friction.
Therefore that force must widen the area of the tires. The wider and bigger the tires are, the bigger is the force against slipping and therefore the faster the racing car accelerates.
During burnout, a melting on the surface can be felt but wide tires neutralize slippery concrete in general. This brief burnout puts some water to the tires to make the tire tacky and also takes out dirt in the surface of the tires. At times, wings are installed to expert fuel dragsters that results in a downward force.
This downward force can go as high as 8000 pounds on a 2000 pound engine when travelling 300 mph. Thus tire slipping is not considered a problem in high acceleration because the tires are made stable to neutralize slipping.