Classification of a family of symmetric graphs with complete 2-arc-transitive quotients

In this paper we give a classification of a family of symmetric graphs with complete 2-arc-transitive quotients. Of particular interest are two subfamilies of graphs which admit an arc-transitive action of a projective linear group. The graphs in these subfamilies can be defined in terms of the cross ratio of certain 4-tuples of elements of a finite projective line, and thus may be called the second type 'cross ratio graphs', which are different from the 'cross ratio graphs' studied in [A. Gardiner, C. E. Praeger, S. Zhou, Cross-ratio graphs, J. London Math. Soc. (2) 64 (2001), 257-272]. We also give a combinatorial characterisation of such second type cross ratio graphs.

[1]  Sanming Zhou,et al.  On a class of finite symmetric graphs , 2008, Eur. J. Comb..

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

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

[4]  C. Lindner,et al.  Design Theory, Second Edition , 2008 .

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

[6]  William M. Kantor,et al.  Homogeneous Designs and Geometric Lattices , 1985, J. Comb. Theory A.

[7]  Sanming Zhou,et al.  Finite symmetric graphs with two-arc transitive quotients , 2005, J. Comb. Theory, Ser. B.

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

[9]  P. Cameron.  FINITE PERMUTATION GROUPS AND FINITE SIMPLE GROUPS , 1981 .

[10]  Sanming Zhou,et al.  Finite symmetric graphs with two-arc transitive quotients , 2005, J. Comb. Theory, Ser. B.

[11]  H. Weyl.  Permutation Groups , 2022 .

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

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

[14]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[15]  Haitze J. Broersma,et al.  Path graphs , 1989, J. Graph Theory.

[16]  Sanming Zhou,et al.  Cross Ratio Graphs , 2001 .

[17]  Sanming Zhou.  Almost Covers Of 2-Arc Transitive Graphs , 2004, Comb..

[18]  Hanfried Lenz,et al.  Design theory , 1985 .

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

[20]  P. M. Neumann,et al.  Groups and Geometry , 1994 .

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

[22]  C. Praeger.  Finite Transitive Permutation Groups and Bipartite Vertex-Transitive Graphs , 2003 .