Finite symmetric graphs with two-arc transitive quotients

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.

[1]  Sanming Zhou,et al.  A class of finite symmetric graphs with 2-arc transitive quotients , 2000, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  Sanming Zhou,et al.  Imprimitive symmetric graphs, 3-arc graphs and 1-designs , 2002, Discret. Math..

[3]  J. Dixon,et al.  Permutation Groups , 1996 .

[4]  Brian Alspach,et al.  A Classification of 2-Arc-Transitive Circulants , 1996 .

[5]  Cheryl E. Praeger,et al.  Remarks on Path-transitivity in Finite Graphs , 1996, Eur. J. Comb..

[6]  Jixiang Meng,et al.  A classification of 2-arc-transitive circulant digraphs , 2000, Discret. Math..

[7]  Cai Heng Li.  A Family of Quasiprimitive 2-arc Transitive Graphs which Have Non-quasiprimitive Full Automorphism Groups , 1998, Eur. J. Comb..

[8]  Sanming Zhou,et al.  Constructing a Class of Symmetric Graphs , 2002, Eur. J. Comb..

[9]  Primoz Potocnik,et al.  On 2-arc-transitive Cayley graphs of Abelian groups , 2002, Discret. Math..

[10]  Cheryl E. Praeger,et al.  Fintte two-arc transitive graphs admitting a suzuki simple group , 1999 .

[11]  Sanming Zhou.  Symmetric Graphs and Flag Graphs , 2003 .

[12]  N. Biggs.  Algebraic Graph Theory: COLOURING PROBLEMS , 1974 .

[13]  W. T. Tutte.  A family of cubical graphs , 1947, Mathematical Proceedings of the Cambridge Philosophical Society.

[14]  Jie Wang,et al.  A Family of Non-quasiprimitive Graphs Admitting a Quasiprimitive 2-arc Transitive Group Action , 1999, Eur. J. Comb..

[15]  Cheryl E. Praeger,et al.  A Geometrical Approach to Imprimitive Graphs , 1995 .

[16]  Cheryl E. Praeger,et al.  Finite two-are transitive graphs admitting a ree simple group , 1999 .

[17]  Cheryl E Praeger.  Surveys in Combinatorics, 1997: Finite Quasiprimitive Graphs , 1997 .

[18]  Cheryl E. Praeger,et al.  A geometric approach to imprimitive symmetric graphs , 1995 .

[19]  Sanming Zhou.  Classifying a family of symmetric graphs , 2001, Bulletin of the Australian Mathematical Society.

[20]  W. T. Tutte.  On the Symmetry of Cubic Graphs , 1959, Canadian Journal of Mathematics.

[21]  Dragan Marusic,et al.  On 2-arc-transitivity of Cayley graphs , 2003, J. Comb. Theory, Ser. B.

[22]  Sanming Zhou.  Almost covers of 2-arc transitive graphs , 2007, Comb..