Imprimitive Symmetric Graphs

A finite graph Γ is said to be G-symmetric if G is a group of automorphisms of Γ acting transitively on the ordered pairs of adjacent vertices of Γ. In most cases, the group G acts imprimitively on the vertices of Γ, that is, the vertex set of Γ admits a nontrivial G-invariant partition B. The purpose of this thesis is to study such graphs, called imprimitive G-symmetric graphs. In the first part of the thesis, we discuss in detail the geometric approach, introduced by Gardiner and Praeger in 1995, for studying imprimitive symmetric graphs which we use throughout. According to this approach, three configurations can be associated with (Γ,B), namely the quotient graph ΓB of Γ with respect to B, the bipartite subgraph Γ[B,C] of Γ induced by two adjacent blocks B,C of B, and a certain 1-design D(B) induced on B (possibly with repeated blocks). The approach involves an analysis of these configurations and addresses the problem of reconstructing Γ from the triple (ΓB,Γ[B,C],D(B)). In the second part, we study the case where the block size k of D(B) is one less than the block size v of B. We first assume that D(B) contains no repeated blocks, and prove that, under the assumption k = v − 1 ≥ 2, this occurs precisely when ΓB is (G, 2)-arc transitive. In this case, we find a very natural and simple construction of Γ from ΓB and the induced action of G on B, and prove that up to isomorphism it produces all such graphs Γ. If in addition ΓB is a complete graph, then we classify all the possibilities for (Γ, G). We show that Γ[B,C] ∼= Kv−1,v−1 if and only if ΓB is (G, 3)-arc transitive, and that Γ[B,C] is a matching of v − 1 edges and ΓB is not a complete graph if and only if ΓB is a certain near n-gonal graph for some even integer n ≥ 4. In the general case where D(B) may contain repeated blocks, we give a construction of such graphs from G-pointand G-block-transitive 1-designs, and prove further that up to isomorphism it gives rise to all such graphs. By using this, we then classify such graphs arising from the classical projective and affine geometries. In the last part, we will investigate the influence of certain “local” actions induced by the setwise stabilizer GB on the structure of Γ, with emphasis on the case where Γ is G-locally quasiprimitive. In particular, we will study the case where the actions of GB on B and on the neighbourhood of B in ΓB are permutationally isomorphic.