# Confusion about the definition of graphs

A graph is a non-empty finite set $V$ of elements called vertices together with a possibly empty set $E$ of pairs of vertices called edges. Here are a few examples of graphs:

1. Vertex set $V = \{a, b, c, d\}$ and edge set $E = \{(a, b), (b, d)\}$
2. Vertex set $V = \{1, 2, 3, 4\}$ and edge set $E = \{(2, 4)\}$
3. Vertex set $V = \{wolf, goat, cabbage\}$ and edge set $E = \{(wolf, cabbage)\}$
4. Vertex set $V = \{A, B, C\}$ and edge set $E = \emptyset$.

Is this the correct definition for graphs ? If the graph has possibly empty empty edges how can it be represented diagramatically ? What can be a practical example of a graph where there are no edges at all ?

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It is weird to not allow empty graphs but to allow at the same time empty edge sets! –  Mariano Suárez-Alvarez Sep 19 '12 at 6:20
It is a correct definition of a simple graph. A graph with no edges can be represented diagrammatically as a finite set of points. Pick just about any application and ask yourself what a graph with no edges would represent; in most cases it will be something meaningful. (And in any case they’re very useful within graph theory.) –  Brian M. Scott Sep 19 '12 at 6:20
$V$ does not have to be finite, in general –  Belgi Sep 19 '12 at 6:22
If you had read a few more sentences in that web lesson, you would have stumbled over a Java Web Start Application that specifically deals with the concept of null graphs, i.e. graphs with $E=\emptyset$. (That's why I downvote). –  Hagen von Eitzen Sep 19 '12 at 6:34
@Hagen: didn't the OP actually copy the definition from that lesson? i imagine the restriction to non-empty vertex sets is not his idea... In that case, downvoting because he is faithful to the source he is getting this from is strange. –  Mariano Suárez-Alvarez Sep 19 '12 at 6:36

## 1 Answer

Just a bunch of vertices with no edges connecting them. Dots.

An example might be islands. Or a collection of social hermits.

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Or a hard-wired network after someone snips the cables. –  Brian M. Scott Sep 19 '12 at 6:22