[Always Right]

What about prayer at graduation that is brought about, prompted, originated or instigated by the government? Is that sort of thing permissible?

If the government mandates it, it is not. If the government simply protects the right of someone to say a prayer, then it is permissible.

[FredFlash]

A mandate carries a punishment for disobeying it, right? So the government can suggest, encourage, facilitate, urge or direct as long as no punishment is involved.