How can I remove duplicates to keep only the row with the lowest index? (Pandas)

Question

I have a CSV file with the following:

    date    running_balance
0   18/09/2017  5803.66
1   18/09/2017  5812.39
2   18/09/2017  6062.39
3   14/09/2017  6099.25
4   13/09/2017  6113.88
5   13/09/2017  6118.88

I want to graph this only using the latest balance information for any particular day. How can I make this simply return:

    date    running_balance
0   18/09/2017  5803.66
3   14/09/2017  6099.25
4   13/09/2017  6113.88

Can I simply keep the rows grouped by date where the index is at a minimum?

Thanks.