Hello,
I am studying the concept of automorphism in graphs and how to look for redundancy in networks by forming automorphism group Aut(G).
A geometric factorization is performed onto the Aut(G)=H1*H2... where Hi represents a symmetric subgroup/subgraph. To understand this factorization I am trying to understand the terms support, support disjoint and generator/generation sets.
Also I am trying to understand the term orbits which in terms of a graph is defined as follws:
vertex v ∈V (G ) is the set Δ(v) = {πv ∈V (G ) : π ∈ Aut(G )}
Does the term πv mean that π ∈ Aut(G ) is permuting the vertex element v ∈V (G )?
Can anyone please provide an example and explanations of the aforementioned concepts in lay-terms?
Hello Dr. Peter Breuer.
Attached are a bunch of slides from my main powerpoint presentation which I am working on pertaining to my research.
This example is from Fraleigh's book on group theory.
There is a square which has a set of vertices V={1,2,3,4}. This square is rotated 90-degree clockwise which can be represented as a bijection
1-->2
2-->3
3-->4
4-->1
on the vertex set V itself. Hence this is called as a permutation.
The definition of a support of permutation that I have encountered says that a support of a permutation p is
supp(p) = {xi | p(xi) not equal to xi}.
Does that mean that a support of a permutation is the same as the permutation itself. Could you please provide an example of a support from this square permutation.
Also an example of orbit from the given permutation group on the vertices of the square?
Hello Dr. Peter Breuer,
A couple of questions from the same example
1. How can I define a generator set based on this square permutation example?
2. What is the exact meaning of a group action? Does that mean that a permutation group permutates the given set? In this context if given a set of vertices of the square V={1,2,3,4} and permutation group G having a set of permutations such as 90,180 degrees clockwise rotation, rotation by the diagonals and by the perpendicular bisectors.
If I select a permutation g such that g∈G and that happened to be 90-degree clockwise rotation. Is this permutation g i.e the 90-degree clockwise rotation a group action on the vertex set V?
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