You found (or were given I suspect) one of the hardest problems in CS. It is called
"Traveling Salesman Problem"[
^]. It is NP-hard but luckily for you one the best investigated problems since it does have many application in real live.
Googling a bit with "Traveling Salesman Problem" or "Traveling Salesman Algorithm" should yield enough results to keep you busy for the next year.
Do some research and you'll surely come up with a solution.
Modification:
Since you didn't introduce any weights (distances, cost) into the edges of the graph, the problem might indeed be easier to solve than the traditional TSP. If you (or your tutor) can accept leaving out loops in the route then termination will even be guaranteed. Something not easily achieved with edges that can carry positive as well as negative weight (distance, cost).
Best regards,