Abstract: | Let Г be a G-symmetric graph admitting a nontrivial G-invariant partition . Let Г be the quotient graph of Г with respect to . For each block B ∊ , the setwise stabiliser GB of B in G induces natural actions on B and on the neighbourhood Г (B) of B in Г . Let G(B) and G[B] be respectively the kernels of these actions. In this paper we study certain “local actions" induced by G(B) and G[B], such as the action of G[B] on B and the action of G(B) on Г (B), and their influence on the structure of Г. Supported by a Discovery Project Grant (DP0558677) from the Australian Research Council and a Melbourne Early Career Researcher Grant from The University of Melbourne. |