There are billions and billions of them.

Monads arising from adjunctons between graphs and sets:

• The monad which takes a graph and turns it into the complete graph on the same vertices.
• The comonad which takes a graph and turns it into the discrete graph on the same vertices. (This example was edited after Andreas Blass made his comment.)

The monad arising from adjunction between graphs and categories (you have to do this right to avoid loops):

• The monad which takes a graph and creates a new one with the same vertices, but connects two vertices iff there is a path between them in the original graph.

Some other random stuff:

• The monad arising from $\pi_0$: it takes a graph and returns the discrete graph whose vertices are the connected components of the original graph.

I have somehow managed to give only monads that are closure operators (the multiplication is an isomorphism). I will let someone else list some other monads.

There are billions and billions of them. But it turns out I originally suggested two non-examples:

• The non-monad which takes a graph and turns it into the complete graph on the same vertices.
• The comonad which takes a graph and turns it into the discrete graph on the same vertices. (This example was edited after Andreas Blass made his comment.)

And two that still seem to be examples:

• The monad which takes a graph and creates a new one with the same vertices, but connects two vertices iff there is a path between them in the original graph.
• The monad arising from $\pi_0$: it takes a graph and returns the discrete graph whose vertices are the connected components of the original graph.

There are billions and billions of them.

Monads arising from adjunctons between graphs and sets:

• The monad which takes a graph and turns it into the complete graph on the same vertices.
• The monad which takes a graph and turns it into the discrete graph on the same vertices.

The monad arising from adjunction between graphs and categories (you have to do this right to avoid loops):

• The monad which takes a graph and creates a new one with the same vertices, but connects two vertices iff there is a path between them in the original graph.

Some other random stuff:

• The monad arising from $\pi_0$: it takes a graph and returns the discrete graph whose vertices are the connected components of the original graph.

I have somehow managed to give only monads that are closure operators (the multiplication is an isomorphism). I will let someone else list some other monads.

There are billions and billions of them.

Monads arising from adjunctons between graphs and sets:

• The monad which takes a graph and turns it into the complete graph on the same vertices.
• The comonad which takes a graph and turns it into the discrete graph on the same vertices. (This example was edited after Andreas Blass made his comment.)

The monad arising from adjunction between graphs and categories (you have to do this right to avoid loops):

• The monad which takes a graph and creates a new one with the same vertices, but connects two vertices iff there is a path between them in the original graph.

Some other random stuff:

• The monad arising from $\pi_0$: it takes a graph and returns the discrete graph whose vertices are the connected components of the original graph.

I have somehow managed to give only monads that are closure operators (the multiplication is an isomorphism). I will let someone else list some other monads.