Types of symmetry for combinatorial graphs

Question

Let $G$ be an undirected, connected graph without loops. Let's call $G$ symmetric iff it has a non-trivial automorphism (that is a permutation $\pi : V(G) \rightarrow V(G) $ – which is not the identity permutation – that preserves the edge relation). Further, let us denote the set of fix points of a permutation $\pi$ with $FP(\pi)$

Let's say $G$ is point-symmetric if the following applies:

• $G$ is symmetric
• Let $\Pi$ be the set of all non-trivial permutations which constitute an automorphism for $G$
• Let $V_p = \bigcap_{\pi \in \Pi} FP(\pi)$ , so with other words all the vertices which can't be swapped with some other vertex for an automorphism
• $V_p$ is non-empty

And let's say $G$ is bi-symmetric if the following holds:

• $G$ is symmetric and $|V(G)|$ is even
• There exists a subset $V_b \subset V(G)$ with $2*|V_b| = |V(G)|$
• The vertex-induced subgraph $G[V_b]$ is connected
• $G[V_b]$ is isomorph to $G[V(G) \setminus V_b]$

Or maybe put more simply, $G$ contains two parts which can be swapped to obtain an automorphism.

I just thought of these two kinds of symmetry for which I think they are described in group theory(maybe as special cases of a more generalized concept) but due to my limited knowledge in this field I wouldn't know where to look. I'm interested in properties of these both types of symmetries.

So my question is: under what kind of name are these concepts of symmetry known or what would be the name of a broader concept containing them?

Answer 1

Well, this was posted some 3 years ago, but I randomly entered it, so I'll try to answer it.

I think that both the symmetries you described, at least in group theory, have no special names or reference, as they are both in some sense trivial.

For example, a Cayley graph of some finite group is never point symmetric, as it is transitive.

The other definition, i.e. bi-symmetric, might have some content to it, since the composition of such automorphism with itself is the identity. Such functions, generally in mathematics, are called 'involution', and they are important in many fields.

Hope this helps.