Asymmetric vs Nonsymmetric Graphs

Question

What are the differences between an asymmetric graph and a nonsymmetric graph?

I think it is:
Asymmetric: No two distinct vertices are symmetric
Nonsymmetric: No complete asymmetry or symmetry for every vertex
Symmetric: Every two distinct vertices are symmetric

Am I correct?

Answer 1

Unfortunately, there are multiple uses of the term symmetric which can cause this sort of confusion. Here are few uses. A symmetric graph could refer to

1. a graph that is both edge- and vertex-transitive;
2. a graph that is arc-transitive;
3. a graph that is undirected. This is your suggestion!

As for the other terms, fortunately they tend to have slightly less ambiguous meaning.

• An asymmetric graph is a graph $G$ such that Aut$(G)=0$, the trivial group. Since automorphisms can be thought of as the symmetry of a group, this term makes some sense.

• An antisymmetric graph is a directed graph where distinct vertices are connected by edges going only in one direction.