Finite two-distance-transitive graphs of valency 6

A non-complete graph Gamma is said to be (G,2)-distance-transitive if, for i = 1,2 and for any two vertex pairs (u_1,v_1) and (u_2,v_2) with d_Gamma(u_1,v_1) = d_Gamma(u_2,v_2) = i, there exists g in G such that (u_1,v_1)^g=(u_2,v_2). This paper classifies the family of (G,2)-distance-transitive graphs of valency 6 which are not (G,2)-arc-transitive.

[1]  Cheryl E. Praeger,et al.  On geodesic transitive graphs , 2015, Discret. Math..

[2]  A. A. Ivanov,et al.  Algebraic, Extremal and Metric Combinatorics, 1986: Distance-Transitive Graphs of Valency k , 8 ≤ k ≤ 13 , 1988 .

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

[4]  John van Bon.  Affine Distance-Transitive Groups , 1993 .

[5]  A. Neumaier,et al.  Distance Regular Graphs , 1989 .

[6]  Cheryl E. Praeger,et al.  DISTANCE TRANSITIVE GRAPHS AND FINITE SIMPLE GROUPS , 1987 .

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

[8]  H. Wielandt,et al.  Finite Permutation Groups , 1964 .

[9]  Cheryl E. Praeger,et al.  Line graphs and geodesic transitivity , 2013, Ars Math. Contemp..

[10]  Richard M. Weiss,et al.  Distance-transitive graphs and generalized polygons , 1985 .

[11]  Cheryl E. Praeger,et al.  On Finite Affine 2-Arc Transitive Graphs , 1993, Eur. J. Comb..

[12]  Cheryl E. Praeger,et al.  On a reduction theorem for finite, bipartite 2-arc-transitive graphs , 1993, Australas. J Comb..

[13]  Arjeh M. Cohen.  Local recognition of graphs, buildings, and related geometries , 1990 .

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

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

[16]  P. Cameron,et al.  PERMUTATION GROUPS , 2019, Group Theory for Physicists.

[17]  Tatsuro Ito,et al.  On distance-regular graphs with fixed valency , 1987, Graphs Comb..

[18]  D. G. Higman.  Intersection matrices for finite permutation groups , 1967 .

[19]  Cai Heng Li,et al.  Finite 2-arc-transitive abelian Cayley graphs , 2008, Eur. J. Comb..

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

[21]  Derek H. Smith.  Distance-Transitive Graphs of Valency Four , 1974 .