A Local Analysis of Imprimitive Symmetric Graphs

Let Г be a G-symmetric graph admitting a nontrivial G-invariant partition $${\cal B}$$. Let Г$$_{\cal B}$$ be the quotient graph of Г with respect to $${\cal B}$$. For each block B ∊ $${\cal B}$$, the setwise stabiliser GB of B in G induces natural actions on B and on the neighbourhood Г$$_{\cal B}$$(B) of B in Г$$_{\cal B}$$. 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$$_{\cal B}$$(B), and their influence on the structure of Г.