# Is there any difference between empty graph and null graph?

I was going through one of the links

Is the empty graph connected?

here I found that it is mentioned that empty graph is a graph with no edges while null graph is a graph with no vertices but I studied on the wikipedia that null graph is a graph which can have both cases:

case 1 : the graph may have no vertices case 2: the graph may have vertices but no edges in it.

so case 2 corresponds to the definition of the empty graph ,so I am confused in this,plz clarify.

## 1 Answer

Terminology sometimes differ between what field of mathematics you come from, or may even differ between mathematicians. In this case, you have found a concept which, depending on who you ask, is well defined proabably as your case 1 or case 2. This is the reason why it is important in a good book to have clear definitions, even of the easy stuff, since it may differ between authors exactly what they mean when they say a concept.

Another example of this is the concept of subgraph. In graph theory it is common that a subgraph may exclude only edges (hence if you remove all edges from a graph, that is a subgraph of the original graph). While if you come from more the logic side of mathematics and talk about subgraphs, you probably mean that a subgraph is just taking a subset of the node set, and keeping all the edges (hence the graph with all edges removed isnt a subgraph if you had edges in the original graph).

The deeper into mathematics you go the more common this gets, since newer concepts often have less established names.

Conclusion: Its just terminology which isn't completely established. Empty graph and null graph are may both be either the graph without any vertices, or a graph with vertices but without any edges. It is hence important for evertone to briefly hint on what they mean when they state talk about the empty graph.

• Hmm, I beg to differ about the concept of subgraph. In graph theory, it is common that a subgraph may exclude vertices as well. For example we can say things like "a connected subgraph". And so I illustrate your point. =) – user21820 Mar 19 '15 at 13:16
• Notice the word 'may' in my answer. Hence I do not exclude the possibility of removing vertices, however in this case, to make the example easy and to show my point, I only remove edges. – Ove Ahlman Mar 19 '15 at 13:30
• So sir ,you said that if I remove all the edges from the graph ,then the resulting graph is a subgraph ,but how is that possible? – radhika Mar 19 '15 at 13:42
• Also ,what conclusion should I derive from this that I can use null and empty graphs interchangebaly or not? – radhika Mar 19 '15 at 13:43
• The point of the subgraph example is that there are multiple definitions of the concept. I've updated my answer now with a more clear conclusion. You should not use both concepts interchangeably, but instead decide what you what to call what, stick with it, and remember to define it when you talk to others so that they know what you mean. – Ove Ahlman Mar 19 '15 at 13:57