AutomorphismsOfConcepts

So automorphisms is a pretty ugly word , but it's not such a hard notion.

Definition An automorphism is a permutation on the vertices (mapping from set of vertices to set of vertices) of a graph that leave the edge relations (set with elements of the form (vertex, vertex)) unchanged.

Here's the picture

a - b   b - a
 \ /     \ /
  c   =   c     you can switch a & b  
  |       |
  d       d

a - b   c - b
 \ /     \ /
  c  !=   a     you can't switch a & c  
  |       |
  d       d

In fact, the set of automorphisms forms a group (under compositon). What that means is that if you take two automorphisms and apply them in any order, the result is also an automorphism (and each one has an inverse and there's an identity).

For a tree, you can always switch the leaves

        root
         |
        / \
       a   b

If a, b are sub-concepts (sub-graphs) then you can move the entire sub-concept too, and that will be an aoutomorphism too.

What is the interpretation of this? You can re-order the outline at will and nothing changes. If this seems contrary to your intuition, it is because the outline does not explicitly show the dependancy of subsequent materials on it's prerequisits (i.e. the concepts are not a tree).

At this point there are three digressions of interest (note their graph structure):

• CaleyColorGraph? and the RubiksCube?
• RamseyTheory? and the RamseyPartyProblem?

It is also worth noting that graphs can be genralized such that the edges are colored (labeled), weighted, or directed.