What is the term for a graph on $n$ vertices with no edges?

Question

What is the term for a graph comprised of $n$ pairwise disconnected vertices?

I could call these $1$-colorable graphs or something like that, but I would rather use standard terminology if it exists.

Also is there a notation to go along with the name? For example $C_n$ generally denotes cycle graphs.

Answer 1

The standard terminology is to call it the empty graph. Let me quote from the text [Bollobas, Modern Graph Theory, p.3], where he calls it the "empty graph": ``As $E_n$ is rather close to the notation for the edge set of a graph, we frequently use $\overline{K}_n$ for the empty graph of order $n$, signifying that it is the complement of the complete graph. In general, for a graph $G=(V,E)$ the complement of $G$ is $\overline{G}=(V,V^{(2)}-E)$; thus, two vertices are adjacent in $\overline{G}$ if and only if they are not adjacent in G.''

Answer 2

According to Wikipedia:

If it is a graph with no edges and any number $n$ of vertices, it may be called the null graph on $n$ vertices. (There is no consistency at all in the literature.)

You didn’t ask about notation, as distinct from terminology, but if the context is simple graphs, I’d denote it by $\langle[n],\varnothing\rangle$ or, if a different vertex set was wanted, $\langle V,\varnothing\rangle$ for the appropriate $V$.

Answer 3

In "Introduction to Graph Theory", Douglas B. West calls a graph with no edges a "trivial graph". If it has no edges and no vertices, it's a "null graph".

There is no standardized terminology in graph theory. For this reason, many authors spend a few extra words precisely defining the terminology they use.