I checked Reinhard Diestel's textbook on Graph Theory. p.2

A graph of order 0 or 1 is called trivial. Sometimes, e.g. to start an induction, trivial graphs can be useful; at other times they form silly counterexamples and become a nuisance. To avoid cluttering the text with non-triviality conditions, we shall mostly treat the trivial graphs, and particularly the empty graph, with generous disregard.

Only nonempty graphs are defined to be connected. (p.9)

I checked Reinhard Diestel's textbook on Graph Theory. p.2

A graph of order 0 or 1 is called trivial. Sometimes, e.g. to start an induction, trivial graphs can be useful; at other times they form silly counterexamples and become a nuisance. To avoid cluttering the text with non-triviality conditions, we shall mostly treat the trivial graphs, and particularly the empty graph, with generous disregard.

Only nonempty graphs are defined to be connected. (p.9)