which preposition is correct to use? On graduation day or at graduation day?
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Normally, referring to a day as a period of time, you want on. That is essentially universal, and applies here as much as anywhere else.
However, we also sometimes use the term "graduation day" to refer to the event. It's more common to just use graduation for that, but if your school, college or university does a lot of things for graduation day, not just the ceremony, then you might want to talk about the whole thing, all the components. In that case, it is not incorrect to refer to something happening at graduation day. Some people will get sniffy about it, and if you're a non-native speaker some people might think you are getting it wrong rather than making a distinction, even if they appreciate the distinction and would use it themselves.
My granddad died on my graduation day.
Very sad, your grandfather died the day that you attended your graduation.
My granddad died at my graduation day.
Very traumatic, your grandfather was at your graduation day event and died while he was there. To avoid possibly being seen as incorrect, you could rephrase:
My granddad died while attending my graduation (day).
We say 'on + day', it doesn't matter if you have one or more words in between.