Finite symmetric graphs with two-arc transitive quotients

  • Authors:

  • Mohammad A. Iranmanesh;Cheryl E. Praeger;Sanming Zhou

  • Affiliations:

  • Department of Mathematics, Yazd University, Yazd, Iran;School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia;Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2005

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Abstract

This paper forms part of a study of 2-arc transitivity for finite imprimitive symmetric graphs, namely for graphs Γ admitting an automorphism group G that is transitive on ordered pairs of adjacent vertices, and leaves invariant a nontrivial vertex partition B. Such a group G is also transitive on the ordered pairs of adjacent vertices of the quotient graph ΓB corresponding to B. If in addition G is transitive on the 2-arcs of Γ (that is, on vertex triples (α, β γ) such that α, β and β, γ are adjacent and α ≠ γ), then G is not in general transitive on the 2-arcs of ΓB, although it does have this property in the special case where B is the orbit set of a vertex-intransitive normal subgroup of G. On the other hand, G is sometimes transitive on the 2-arcs of ΓB even if it is not transitive on the 2-arcs of Γ. We study conditions under which G is transitive on the 2-arcs of ΓB. Our conditions relate to the structure of the bipartite graph induced on B ∪ C for adjacent blocks B, C of B and a graph structure induced on B. We obtain necessary and sufficient conditions for G to be transitive on the 2-arcs of one or both of Γ, ΓB, for certain families of imprimitive symmetric graphs.