The equilibrium geometry and electronic structure of graphene on the most stable In-vacancy InAs(111)A surface has been investigated using the density functional and pseudopotential theories. The equilibrium distance between graphene and InAs(111) is found to be 3.05 with adsorption energy approximately 38meV/C atom. According to our electronic band calculation, there is a re-distribution of the charge density around the graphene sheet, which leads to the development of a dipole moment along the surface normal. Scanning tunnelling microscopy image simulations suggest that the InAs(111) substrate is visible through the graphene layer. {\textcopyright} 2012 IOP Publishing Ltd.