I checked Reinhard Diestel's textbook on <a href="http://diestel-graph-theory.com/">Graph Theory</a>. 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)