A Geometrical Approach to Imprimitive Graphs

Abstract

We establish a geometrical framework for the study of imprimitive, G‐symmetric graphs Г by exploiting the fact that any G‐partition B of the vertex set VГ gives rise both to a quotient graph ГB and to a tactical configuration D(B) induced on each block B ∈ B. We also examine those cases in which D(B) is degenerate, and characterize the possible graphs Г in many cases where the quotient ГB is either a complete graph or a circuit. When D(B) is non‐degenerate, a natural extremal case occurs when D(B) is a symmetric 2‐design with stabilizer G(B) acting doubly transitively on points: we characterize such graphs in the case where ГB is complete.

A Local Analysis of Imprimitive Symmetric Graphs

“…One can check that D(B) is a 1-(v, k, r ) design with b blocks and, up to isomorphism, is independent of the choice of B. Also, the setwise stabiliser G B := {g ∈ G : B g = B} of B in G induces a group of automorphisms of D(B), and G B is transitive on the points, the blocks and the flags of D(B) [3]. Thus, the number of times a block C of D(B) is repeated is independent of the choice of B and C. We denote this number by m and call it the multiplicity of D(B).…”

“…Such graphs were studied initially in [8,9], and more recent results were obtained in [6]. The following theorem is a generalization of [3,Lemma 3.4], where is required to be G-locally primitive (that is, G α is primitive on (α)). ] ; in this case G (B) is transitive on (α) for each α ∈ B, and moreover either Now we suppose G [B] ≤ G (B) .…”

“…Gardiner and Praeger [3] suggested that we may analyse the triple ( B , [B, C], D(B)) in order to study . This approach is a geometric one in the sense that it involves the "cross-sectional" geometry D(B).…”

“…To depict genuinely the structure of we also need a "cross-sectional" geometry [3], namely the incidence structure…”

“…Nevertheless, B does not determine completely since it does not tell us how adjacent blocks of B are joined by edges of . To compensate for this shortage, we need [3] the "inter-block" subgraph induced by two adjacent blocks of B. Let (α) := {β ∈ V ( ) : {α, β} ∈ E( )}, the neighbourhood of α in .…”