TITLE: Is a graph with 0 nodes a graph?
QUESTION [6 upvotes]: If you have a graph with 1 node and no edges and you then remove this 1 node, is the remaining a graph with 0 nodes or is it not anymore a graph?
Alternative formulation:
Is there something such as a empty null graph?

REPLY [12 votes]: This is not quite clear. There is an entire paper dedicated to this question.
'Is the null-graph a pointless concept?' by Frank Harary and Ronald C. Read 
Its abstract says (my emphasis):

The graph with no points and no lines is discussed critically. Arguments for and against its official admittance as a graph are presented. This is accompanied by an extensive survey of the literature. Paradoxical properties of the null-graph are noted. No conclusion is reached.