TY - JOUR

T1 - Vertex-imprimitive symmetric graphs with exactly one edge between any two distinct blocks

AU - Fang, Teng

AU - Fang, Xin Gui

AU - Xia, Binzhou

AU - Zhou, Sanming

PY - 2017/11/1

Y1 - 2017/11/1

N2 - A graph Γ is called G-symmetric if it admits G as a group of automorphisms acting transitively on the set of ordered pairs of adjacent vertices. We give a classification of G-symmetric graphs Γ with V(Γ) admitting a nontrivial G-invariant partition B such that there is exactly one edge of Γ between any two distinct blocks of B. This is achieved by giving a classification of (G,2)-point-transitive and G-block-transitive designs D together with G-orbits Ω on the flag set of D such that Gσ,L is transitive on L∖{σ} and L∩N={σ} for distinct (σ,L),(σ,N)∈Ω, where Gσ,L is the setwise stabilizer of L in the stabilizer Gσ of σ in G. Along the way we determine all imprimitive blocks of Gσ on V∖{σ} for every 2-transitive group G on a set V, where σ∈V.

AB - A graph Γ is called G-symmetric if it admits G as a group of automorphisms acting transitively on the set of ordered pairs of adjacent vertices. We give a classification of G-symmetric graphs Γ with V(Γ) admitting a nontrivial G-invariant partition B such that there is exactly one edge of Γ between any two distinct blocks of B. This is achieved by giving a classification of (G,2)-point-transitive and G-block-transitive designs D together with G-orbits Ω on the flag set of D such that Gσ,L is transitive on L∖{σ} and L∩N={σ} for distinct (σ,L),(σ,N)∈Ω, where Gσ,L is the setwise stabilizer of L in the stabilizer Gσ of σ in G. Along the way we determine all imprimitive blocks of Gσ on V∖{σ} for every 2-transitive group G on a set V, where σ∈V.

KW - arc-Transitive graph

KW - Flag graph

KW - Spread

KW - Symmetric graph

UR - http://www.scopus.com/inward/record.url?scp=85021292329&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2017.06.007

DO - 10.1016/j.jcta.2017.06.007

M3 - Article

AN - SCOPUS:85021292329

SN - 0097-3165

VL - 152

SP - 303

EP - 340

JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

ER -