A classification of the m-graphical regular representation of finite groups

In this paper we extend the classical notion of digraphical and graphical regular representation of a group and we classify, by means of an explicit description, the finite groups satisfying this generalization. A graph or digraph is called regular if each vertex has the same valency, or, the same out-valency and the same in-valency, respectively. An m-(di)graphical regular representation (respectively, m-GRR and m-DRR, for short) of a group G is a regular (di)graph whose automorphism group is isomorphic to G and acts semiregularly on the vertex set with m orbits. When m=1, this definition agrees with the classical notion of GRR and DRR. Finite groups admitting a 1-DRR were classified by Babai in 1980, and the analogue classification of finite groups admitting a 1-GRR was completed by Godsil in 1981. Pivoting on these two results in this paper we classify finite groups admitting an m-GRR or an m-DRR, for arbitrary positive integers m. For instance, we prove that every non-identity finite group admits an m-GRR, for every m>4.

[1]  D. Robinson.  A Course in the Theory of Groups , 1982 .

[2]  Steve Wilson,et al.  Characterization of edge‐transitive 4‐valent bicirculants , 2012, J. Graph Theory.

[3]  W. Imrich,et al.  Tournaments with given regular group , 1978 .

[4]  Pablo Spiga,et al.  Finite groups admitting an oriented regular representation , 2018, J. Comb. Theory, Ser. A.

[5]  M. Watkins,et al.  On the action of non-Abelian groups on graphs , 1971 .

[6]  Wilfried Imrich.  On products of graphs and regular groups , 1972 .

[7]  Joy Morris,et al.  Every finite non-solvable group admits an oriented regular representation , 2017, J. Comb. Theory, Ser. B.

[8]  Gerd Baron,et al.  Asymmetrische reguläre Graphen , 1969 .

[9]  P. Spiga,et al.  Classification of finite groups that admit an oriented regular representation , 2017, Bulletin of the London Mathematical Society.

[10]  G. Sabidussi.  Vertex-transitive graphs , 1964 .

[11]  W. Imrich,et al.  On automorphism groups of Cayley graphs , 1976 .

[12]  Dragan Marusic,et al.  On strongly regular bicirculants , 2007, Eur. J. Comb..

[13]  Dragan Marusic,et al.  On normality of n-Cayley graphs , 2018, Appl. Math. Comput..

[14]  Yan-Quan Feng,et al.  The automorphisms of bi-Cayley graphs , 2016, J. Comb. Theory, Ser. B.

[15]  Joy Morris,et al.  Most rigid representations and Cayley index , 2017, Art Discret. Appl. Math..

[16]  Yan-Quan Feng,et al.  Cubic bi-Cayley graphs over abelian groups , 2014, Eur. J. Comb..

[17]  C. Godsil.  Tournaments with prescribed regular automorphism group , 1986 .

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

[19]  P. Spiga,et al.  Asymptotic enumeration of Cayley digraphs , 2018, Israel Journal of Mathematics.

[20]  Dragan Marusic,et al.  Strongly regular tri-Cayley graphs , 2009, Eur. J. Comb..

[21]  Luis Martínez,et al.  Partial sum quadruples and bi-Abelian digraphs , 2012, J. Comb. Theory, Ser. A.

[22]  Binzhou Xia,et al.  Cubic graphical regular representations of , 2016, Discret. Math..

[23]  László Babai,et al.  Finite digraphs with given regular automorphism groups , 1980 .

[24]  Wilfried Imrich,et al.  On graphical regular representations of cyclic extensions of groups , 1974 .

[25]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[26]  Dragan Marusic,et al.  One-matching bi-Cayley graphs over abelian groups , 2009, Eur. J. Comb..

[27]  Gert Sabidussi,et al.  Graph multiplication , 1959 .