Finite symmetric graphs with two-arc transitive quotients II

• Authors: Zaiping Lu;Sanming Zhou
• Venue: Journal of Graph Theory
• Year: 2007

Let Γ be a finite G-symmetric graph whosevertex set admits a nontrivial G-invariant partitionB. It was observed that the quotient graphΓB of Γ relative toB can be (G, 2)-arc transitive even ifΓ itself is not necessarily (G, 2)-arctransitive. In a previous article of Iranmanesh et al., thisobservation motivated a study of G-symmetric graphs(Γ,B) such that ΓB is(G, 2)-arc transitive and, for blocks B,C εB adjacent inΓB, there are exactly |B| - 2(≥1)vertices in B which have neighbors in C. In thepresent article we investigate the general case whereΓB is (G, 2)-arc transitive and isnot multicovered by Γ (i.e., at least one vertex inB has no neighbor in C for adjacent B, CεB) by analyzing the dual D*(B)of the 1-designD(B):=(B,ΓB(B),I),where ΓB(B) is theneighborhood of B in ΓB andαIC (α ε B, C εΓB(B)* inD(B) if and only if α has at leastone neighbor in C. In this case, a crucial feature is thatD*(B) admits G as a group of automorphismsacting 2-transitively on points and transitively on blocks andflags. It is proved that the case when no point ofD(B) is incident with two blocks can be reduced tomulticovers, and the case when no point of D(B) is incident withtwo blocks can be partially reduced to the 3-arc graphconstruction, where D(B) is the complementof D(B). In the general situation, bothD*(B) and its complement D* are (G,2)-point-transitive and G-block-transitive 2-designs, andexploring relationships between them and Γ is anattractive research direction. In the article we investigate thedegenerate case where D*(B) or D* is a trivial Steinersystem with block size 2, that is, a complete graph. In each ofthese cases, we give a construction which produces symmetric graphswith the corresponding properties, and we prove further that everysuch graph Γ can be constructed fromΓB by using the construction. © 2007Wiley Periodicals, Inc. J Graph Theory 56: 167193, 2007