  Let F be a finite field and R = M2(F) be 2x2 matrix ring over F. In this
paper, we explicitly determine all the idempotents in R. Using these
idempotents, we study the idempotent graph of R whose vertex set is the set of
non-trivial idempotents in R and two idempotents e, f are adjacent if ef = 0 or
fe = 0. It is proved that the idempotent graph of R is connected regular graph
with diameter 2. Its girth is also characterized. Further, we determine the
Wiener and Harary index of the idempotent graph of R.
