Different layer of graphene have different band gap. for example Single layer has no band gap, but bilayer has band gap, can be used to make transistor. I don't know much about multilayer. Band gap is basic properties of any material. So according to me work function should be different for different layer.

This answer base on my previous work memory.

Which one is better to make transistor? High or low value of work function?

You at least need two layers of graphene to build a field-effect tunneling transistor... Normally due to the very small band gap, graphene can not be used solely, you need some other materials as transport barriers between the layers, and the materials you use will determine the work function, from that point conventional rules apply.

You should have a look at this patent:

http://www.google.com.tr/patents?hl=tr&lr=&vid=USPAT7732859&id=5m7RAAAAEBAJ&oi=fnd&dq=graphene+for+transistors&printsec=abstract#v=onepage&q=graphene%20for%20transistors&f=false

You can also consider that the graphene fabrication process is determinant in the device properties. CVD graphene can show different aspects comparing to exfoliated graphene. I suggest you use CVD grown graphene for better results. Good luck!

You can use multipol graphene layer which a decrease in resistivity but the transparency will decrease.

Yes, work function of graphene depends on number of layers (http://prb.aps.org/pdf/PRB/v83/i23/e235434). To fabricate high performance graphene transistor is a personal choice i.e. one want high mobility/frequency operation or higher on/off ratio. For the former requirement single layer is best for later bilayer since it has a band gap.

Generally, work function of graphene can be perturbed easily using add-layers such as metal nanoparticles, oxide nanoparticles, SAM,etc. So according to me work function can not be considered as bottleneck property, in comparison to band gap, for graphene transistor. I hope this reply has answered your question.