First of all I am sorry to include an image but am not sure how to post this otherwise.
I am having some trouble understanding what constitutes an automorphsim of a graph. let's consider the top left tree and mark its vertices starting from the upper leaf and continuing clockwise (with the graph) as ${1, 2, 3, 4, 5, 6}$. so according to this solution the only automorphisms are $\sigma(i) = i $, and $\sigma(1) = 6, \sigma(6) = 1, \sigma(i)_{2 \leq i \leq 5} = i$.
Why isn't $\sigma(3) = 4, \sigma(4) = 3, \sigma(i)_{\text{else}} = i$ also an automorphsim? it holds $(i, j) \in E$ iff $ (\sigma(i), \sigma(j))\in E_{new}$, what am I missing?
