graph automorphism

Let $G=(V,E)$ be a graph. A graph automorphism is a graph homomorphism from $G$ to $G$. That is, a map $\gamma :G\to G$ such that

If there exists an automorphism for $G$, then there exists a corresponding permutation matrix $P$ such that $V=PV$.

In a learning task (in a GNN), this means that the shift operator $S=PS{P}^T$ and node features $x=Px$ for the permutation matrix $P$.