I have a heavily symmetric regular graph whose automorphisms I know. I remove one subgraph and insert another one in a consistent manner; for example, this could be a Delta-Y transformation (replacing a node with a complete subgraph). I'd like to compute the automorphisms of the new graph using the automorphisms of the old graph.


I have the sense that this problem isn't too difficult, but am not sure how best to approach it and am new to algebraic graph theory. 


What are some good references or suggested starting points?



How are graph automorphisms are affected by transformations?
I do not think there are any results that relate the automorphism group of a graph after
subgraph replacement to the group of the original graph. 


If the original graph was "heavily symmetric", you would expect the group of the new graph to be smaller - most local operations would destroy vertex transitivity, for example - but the details would depend on the operation. 


I have the sense that this problem is extremely difficult. There are very few results
in algebraic graph theory which determine the full automorphism group of a graph.
There are results that describe the automorphism group of a graph product in terms of the
automorphism groups of its factors. (See "Products Graphs" by Imrich and Klavsar, for example.) 


