Is there a nodeless graph?

Solution 1:

Depends on your definition of a graph. This is addressed in a paper of Harary, Is the null-graph a pointless concept? The abstract is as follows:

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.

Personally, I think there's no reason not to admit it as a graph. There is a philosophy due to Grothendieck that it's better to work in a nice category with nasty objects than in a nasty category with nice objects, and the null graph makes the category of graphs nicer (giving it an initial object). As for the paradoxical properties, I haven't read the paper but they are likely due to the phenomenon that the nLab calls too simple to be simple. For example, the empty graph is not connected; it has zero connected components instead of one.