The easiest way to do that would be to use an external pulldown resistor (a resistor from the reset pin to ground) and connect to Vcc (5V on the Nano) via your alligator cables.
As long as the alligator cables are connected, the reset pin is held HIGH through them. When you disconnect them, the pulldown resistor pulls the reset pin to LOW. That holds the Nano in reset until you again connect the alligator clips.
If you want the Nano to reset and then directly work again without the alligator clips reconnected, it gets more difficult. One way I can think of is building a monostable vibrator circuit, which gets in the excited state, when your alligator clips are disconnected, resets the Arduino and then goes back into the stable stable, letting the Arduino execute its code. Though that is more an electrical question, than an Arduino one.