In this paper we extend the notion of oriented regular representations in the context of m-partite oriented digraphs with an integer m≥2. A group G is said to admit an m-partite oriented semiregular ...representation (m-POSR for short) if there exists an m-partite oriented digraph such that its automorphism group is isomorphic to G and acts semiregularly on vertices with every part of the digraph as an orbit. In this paper, we proved that G admits an m-POSR of valency two except for several cases when G is a finite group generated by at most two elements and m≥2 is an integer.
Given a group G, an m-Cayley digraph Γ over G is a digraph that has a group of automorphisms isomorphic to G acting semiregularly on the vertex set with m orbits. We say that G admits an oriented ...m-semiregular representation (OmSR for short), if there exists a regular m-Cayley digraph Γ over G such that Γ is oriented and its automorphism group is isomorphic to G. In particular, O1SR is also named as ORR. Verret and Xia gave a classification of finite simple groups admitting an ORR of valency two in Verret and Xia (2022) 19. Let m≥2 be an integer. In this paper, we show that all finite groups generated by at most two elements admit an OmSR of valency two except four groups of small orders. Consequently, a classification of finite simple groups admitting an OmSR of valency two is obtained.
A group G admits an n-partite digraphical representation if there exists a regular n-partite digraph Γ such that the automorphism group Aut(Γ) of Γ satisfies the following properties:(1)Aut(Γ) is ...isomorphic to G,(2)Aut(Γ) acts semiregularly on the vertices of Γ and(3)the orbits of Aut(Γ) on the vertex set of Γ form a partition into n parts giving a structure of n-partite digraph to Γ.
In this paper, for every positive integer n, we classify the finite groups admitting an n-partite digraphical representation.
In this paper we extend the notion of digraphical regular representations in the context of Haar digraphs. Given a group G, a Haar digraphΓ over G is a bipartite digraph having a bipartition {X,Y} ...such that G is a group of automorphisms of Γ acting regularly on X and on Y. We say that G admits a Haar digraphical representation (HDR for short), if there exists a Haar digraph over G such that its automorphism group is isomorphic to G. In this paper, we classify finite groups admitting an HDR.
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. A (di)graphical m-semiregular representation (respectively, GmSR and DmSR, 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 D1SR were classified by Babai in 1980, and the analogue classification of finite groups admitting a G1SR was completed by Godsil in 1981. Pivoting on these two results in this paper we classify finite groups admitting a GmSR or a DmSR, for arbitrary positive integers m. For instance, we prove that every non-identity finite group admits a GmSR, for every m≥5.
On normality of n-Cayley graphs Hujdurović, Ademir; Kutnar, Klavdija; Marušič, Dragan
Applied mathematics and computation,
09/2018, Letnik:
332
Journal Article
Recenzirano
Let G be a finite group and X a (di)graph. If there exists a semiregular subgroup G¯ of the automorphism group Aut(X) isomorphic to G with n orbits on V(X) then the (di)graph X is called an n-Cayley ...graph on G. If, in addition, this subgroup G¯ is normal in Aut(X) then X is called a normal n-Cayley graph on G.
In this paper the normalizers of semiregular subgroups of the automorphism group of a digraph are characterized. It is proved that every finite group admits a vertex-transitive normal n-Cayley graph for every n ≥ 2. For the most part the graphs are constructed as Cartesian product of graphs. It is proved that a Cartesian product of two relatively prime graphs is Cayley (resp. normal Cayley) if and only if the factor graphs are Cayley (resp. normal Cayley). In addition, the concept of graphical regular representations (GRRs) is generalized to n-GRR in a natural way, and it is proved that any group admitting a GRR also admits an n-GRR for any n ≥ 1.
A graph-theoretic environment is used to study the connection between imprimitivity and semiregularity, two concepts arising naturally in the context of permutation groups. Among other, it is shown ...that a connected arc-transitive graph admitting a nontrivial automorphism with two orbits of odd length, together with an imprimitivity block system consisting of blocks of size 2, orthogonal to these two orbits, is either the canonical double cover of an arc-transitive circulant or the wreath product of an arc-transitive circulant with the empty graph
K
¯
2
on two vertices.
The following problem is considered: if
H
is a semiregular abelian subgroup of a transitive permutation group
G
acting on a finite set
X
, find conditions for (non)existence of
G
-invariant ...partitions of
X
. Conditions presented in this paper are derived by studying spectral properties of associated
G
-invariant digraphs. As an essential tool, irreducible complex characters of
H
are used. Questions of this kind arise naturally when classifying combinatorial objects which enjoy a certain degree of symmetry. As an illustration, a new and short proof of an old result of Frucht et al. (Proc Camb Philos Soc 70:211–218,
1971
) classifying edge-transitive generalized Petersen graphs, is given.
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