The word "alaverdi" is recorded only in spelling dictionaries. In the explanatory dictionaries, as far as I know, it is not. Nevertheless, I will first tell you about the etymology of the word. It comes from the Arabic "allah" - "god" and the Turkish verb "ver" - "to give" (past tense - "verdi"). Thus, alaverdi literally translates into Russian as "God gave". As for the meaning of this word, it is directly related to the feast.
According to the tradition of the classic Georgian feast, it is entrusted to the toastmaster from beginning to end. At the table he is a king and a god. He, having uttered another toast, can decide to continue it by one of the participants in the feast. In this case, the "relay race" is passed by the toastmaster with the help of the magic word "alaverdi".
Well, one more of the meanings is a response, as a token of gratitude. In the narrow sense - "mutually".
Let's say they gave you something, and you give something in return. Alaverdy!
Now you know that "alaverdi" means that "the floor is given to comrade Saa..khov." :) or "from our table to yours" depending on the situation.