Josephson effect in mesoscopic graphene strips with finite width

We study Josephson effect in a ballistic graphene strip of length L smaller than the superconducting coherence length and arbitrary width W. We find that the dependence of the critical supercurrent I-c on W is drastically different for different types of the edges. For smooth and armchair edges at low concentration of the carriers I-c decreases monotonically with decreasing W/L and tends to a constant minimum for a narrow strip W/L less than or similar to 1. The minimum supercurrent is zero for smooth edges but has a finite value e Delta(0)/h for the armchair edges. At higher concentration of the carriers, in addition to this overall monotonic variation, the critical current undergoes a series of peaks with varying W. On the other hand in a strip with zigzag edges the supercurrent is half-integer quantized to (n+1/2)4e Delta(0)/h, showing a stepwise variation with W.