I've been trying to formalise the following idea in graphs: I want to say that two nodes are "similar" if the rest of the graph looks the same from their point of view. For example, in a cycle every node is in the same class, but a star has two classes: all the leaf nodes are similar to each other but distinct from the centre. Similarly, an extended star has 3 classes.
Here's my attempt at the definition: "nodes $u$ and $v$ are [similar] in $G=(V,E)$ if there exists an automorphism $\varphi: V \rightarrow V$ such that $\varphi(u) = v$." This seems well-defined and seems to capture the intuition correctly.
My question is does this concept have a name in graph theory or group theory? It seems very natural to me but I couldn't find anything. If my definition doesn't work, is there a reasonable definition that captures the notion of "similarity" described above?
That is how "similar nodes" are defined in Frank Harary, Graph Theory, 1969, and G. Chartrand and L. Lesniak, Graphs & Digraphs, Third Edition, 1996, except that Harary calls nodes "points" while Chartrand & Lesniak call them "vertices".
In my opinion your definition is the right one, and "similar" is the right word.