Finite Symmetric Graphs With 2-Arc-Transitive Quotients: Affine Case

Abstract

Let G be a finite group and Γ a G-symmetric graph. Suppose that G is imprimitive on V(Γ) with B a block of imprimitivity and B := {B g ; g ∈ G} a system of imprimitivity of G on V(Γ). Define Γ B to be the graph with vertex set B such that two blocks B, C ∈ B are adjacent if and only if there exists at least one edge of Γ 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 Γ B is 2-arc-transitive. In this paper, we completely settle one of the cases defined by certain parameters connected to Γ and B and show that there is a unique graph corresponding to this case.2010 Mathematics subject classification: primary 05C25; secondary 05E18.

On Hamiltonian Decomposition Problem of 3-Arc Graphs

“…It was also exploited in investigating several classes of symmetric graphs. See [ 4 , 5 ] and references therein.…”