Premier League football rivalries - biggest rivals by teams

What are the biggest rivalties in English Premier League football ? Who are considered as the biggest rivals for top teams ? What are traditional derby matches in Premier League ?

Big rivalries in English Premier League are:
Arsenal - Tottenham
Arsenal - Chelsea
Chelsea - Tottenham
Manchester United - Liverpool
Manchester United - Manchester City
Manchester United - Arsenal
Liverpool - Everton
Brighton & Hove Albion - Crystal Palace