Why not buy a 12v regulated wall wart, and power the Arduino from it as well as the lights, through either barrel jack or the Vin pin? That way, you don't need to mess around with regulators.
Your schematic isn't right. You need to connect the -ve side of the 15v input to Arduino ground and the source terminal of the mosfet (and the ground terminal of any regulators you may be using). The resistor goes between the mosfet gate terminal and the Arduino output pin.
because the arduinos regulator can not supply enough current to run the LEDs and it would over heat