Let X be a finite connected graph and let G be a vertex-transitive group of automorphisms of X. The pair (X, G) is locally-L if the group induced by the action of the stabiliser Gv on the neighbourhood of a vertex v is permutation isomorphic to L. Using this language, a classical theorem of Tutte states that for locally-A3 and locally-S3 pairs, |G| grows linearly with |V(X)|. More generally, given a transitive permutation group L, we are interested in determining the growth of |G| as a function of |V(X)| for locally-L pairs (X,G). We present new results on this topic and highlight an exciting connection with the study of eigenspaces of graphs over finite fields.