FINITE SYMMETRIC GRAPHS WITH 2-ARC-TRANSITIVE QUOTIENTS: AFFINE CASE

Let G be a finite group and Gamma a G-symmetric graph. Suppose that G is imprimitive on V(Gamma) with B a block of imprimitivity and B : = {B-g; g is an element of G} a system of imprimitivity of G on V(Gamma). Define Gamma(B) to be the graph with vertex set B such that two blocks B; C is an element of B are adjacent if and only if there exists at least one edge of Gamma joining a vertex in B and a vertex in C. Xu and Zhou ['Symmetric graphs with 2-arc-transitive quotients', J. Aust. Math. Soc. 96 (2014), 275-288] obtained necessary conditions under which the graph Gamma(B) is 2-arc-transitive. In this paper, we completely settle one of the cases defined by certain parameters connected to Gamma and B and show that there is a unique graph corresponding to this case.