What is graph ($\emptyset,\emptyset$)?

Question

In certain sense, empty graph is defined as a graph with no edges. So, an empty graph may contain any number of vertices. Further, if a graph with V set of vertices and E set of edges is denoted by (V, E) , then ($\emptyset, \emptyset$) corresponds to an empty graph. Now my question is what ($\emptyset, \emptyset$) actually mean? Does it mean a graph with no edges and no vertices which doesn't sound to have any sense. So, can we use ($\infty, \emptyset$) to represent an empty graph with no edges but may or may not contain any number of vertices? Since i am new to this group, my question may not be appropriate to the group. If so, guide me and clarify my question please.

Answer 1

In graph theory, terminology often differs among different graph theorists so you may even see different definitions in different books (I see this while I was trying to learn the terminology "path", "trail", "cycle",etc.). So you can go with whichever is easier for you to understand and if you are a student in highschool or university, I suggest you to go with the definitions that your teacher gives.

In this case as I see, since empty graph is used for the graphs that have no edges and null graph can be used for a particular case where number of vertices is zero (source: http://mathworld.wolfram.com/EmptyGraph.html), in order to prevent confusion I would suggest you to distinguish them as

• Empty Graph: A graph with no edges generally (So $(\infty, \emptyset)$ is here).

• Null Graph: A graph with no edges and no vertices (So $(\emptyset, \emptyset)$ is here).

But as I said earlier, this can vary from book to book or person to person so if you see a different definition, you may try to adapt it or while solving a problem, you can specify what are the definitions you are using. If you are a self-learner, this doesn't hurt you in my opinion. Otherwise if you are a student who is taking a course including graph theory, I suggest you to follow the definitions that your teacher gives for now, and then you can accept whatever definition you want as long as that definition exists.