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?
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4$\begingroup$ Not all favor allowing an empty graph (no vertices). mathworld.wolfram.com/NullGraph.html $\endgroup$– paw88789Feb 23, 2015 at 17:33
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$\begingroup$ And that mathworld article gives the game away. As with any edge-case of a definition you're free to use any definition you like, and typically you'll use whatever definition results in you not needing language in your favourite theorem(s) to treat the edge case specially. $\endgroup$– Steve JessopFeb 23, 2015 at 23:37
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1$\begingroup$ The graph with no vertices should be called the null graph. Calling it the "empty graph" is confusing, because the term empty graph is commonly applied to any graph with no edges, i.e., the complement of a complete graph. $\endgroup$– bofFeb 23, 2015 at 23:58
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3 Answers
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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.
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3$\begingroup$ It definitely exists, it is depicted on the figure 1 of the paper. $\endgroup$– DavidmhFeb 23, 2015 at 22:47
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2$\begingroup$ Yes! :-) More seriously, such an object exists; this is clear. The question asked, and discussed in the paper, is should it be called/considered a graph. The question is a bit like if $0$ is natural number, some think yes some think no while there is agreement that $0$ exists. $\endgroup$– quid ♦Feb 23, 2015 at 22:57
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3$\begingroup$ I'd say that any concept you can get a whole paper out of, isn't pointless. But ofc the pun in the title is probably sufficient to publish regardless of content. $\endgroup$ Feb 23, 2015 at 23:34
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Yes, there is such a thing. A graph with zero nodes is generally referred to as the null graph. The term empty graph usually refers to a graph with no edges (but possibly some nodes).
Googling these terms will yield some non-trivial results.
answered Feb 23, 2015 at 17:27
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An empty graph might or might not have $0$ nodes. What you are talking about is the empty graph with $0$ nodes. It does exist.