On two-sided group digraphs and graphs

Journal of Combinatorial Theory, Series A, 1999

For a finite group G and a subset S of G which does not contain the identity of G, denote by Cay(G, S) the Cayley digraph of G with respect to S. An automorphism _ of the group G induces a graph isomorphism from Cay(G, S) to Cay(G, S _). In this paper, we investigate groups G and Cayley digraphs Cay(G, S) of G for which the following condition holds: for any T/G, Cay(G, S)$Cay(G, T) if and only if S _ =T for some _ # Aut(G). For a positive integer m, a group G is called an m-DCI-group if the condition holds for all Cayley digraphs of valency at most m; while G is called a connected m-DCI-group if it holds for all connected digraphs of valency at most m. This paper contributes towards a complete classification of finite m-DCI-groups for m 2. It was previously proved by C. H. Li et al. (1998, J. Combin. Theory Ser. B 74, 164 183) that finite m-DCIgroups for m 2 belong to an explicitly determined list DCI(m) of groups. However, it is still an open problem to determine which members of DCI(m) are really m-DCI-groups. We reduce this problem to the problem of determining whether all subgroups of groups in DCI(m) are connected m-DCI-groups. Then we give a complete classification of finite 2-DCI-groups.

Houston journal of mathematics

A. V. Kelarev and C. E. Praeger in [11] gave necessary and sufficient conditions for Cayley graphs of semigroups to be vertex-transitive. Also S. Fan and Y. Zeng in [4] gave a description of all vertex-transitive Cayley graphs of finite bands. In this paper we give similar descriptions for all vertex-transitive Cayley graphs of left groups. Also we extend some of the results to every direct product of a group and a band.

Journal of the Indonesian Mathematical Society, 2011

We call a Cayley digraph 􀀀=Cay(G; S) normal for G if R(G), the rightregular representation of G, is a normal subgroup of the full automorphism groupAut(􀀀) of 􀀀. In this paper we determine the normality of Cayley digraphs of valency2 on the groups of order pq and also on non-abelian nite groups G such that everyproper subgroup of G is abelian.DOI : http://dx.doi.org/10.22342/jims.17.2.3.67-72

Bulletin of The Iranian Mathematical Society, 2014

On the planarity of a graph related to the join of subgroups of a finite group .

Journal of the Iranian Mathematical Society, 2025

In this survey the structure of one of the powerful group presentations, introduced by Alain Bretto is studied which is called G-graph. This survey contains some sections of properties, examples, characterization and groups automorphism of G-graphs. There are also some well-known graphs that have the structure of G-graphs. Moreover, in this survey we compare G-graph with the Cayley graph especially to examine the Hamiltonian property of such graphs for a given group. Furthermore by using the construction of a hypergraph, the spectra of the G-graph is reported. Finally, in the last section, the structure of the G-graph is considered as a network and connectivity property is studied.

International Journal of Applied Sciences and Smart Technologies, 2021

Let G be a fnite group with the set of subgroups of G denoted by S(G), then the subgroup graphs of G denoted by T(G) is a graph which set of vertices is S(G) such that two vertices H, K in S(G) (H not equal to K)are adjacent if either H is a subgroup of K or K is a subgroup of H. In this paper, we introduce the Subgroup graphs T associated with G. We investigate some algebraic properties and combinatorial structures of Subgroup graph T(G) and obtain that the subgroup graph T(G) of G is never bipartite. Further, we show isomorphism and homomorphism of the Subgroup graphs of finite groups. Let be a finite group with the set of subgroups of denoted by , then the subgroup graphs of denoted by is a graph which set of vertices is such that two vertices , are adjacent if either is a subgroup of or is a subgroup of . In this paper, we introduce the Subgroup graphs associated with . We investigate some algebraic properties and combinatorial structures of Subgroup graph and obtain that the su...

Graphs and Combinatorics, 2019

In this paper, we define meta-Cayley graphs on dihedral groups. We fully determine the automorphism groups of the constructed graphs in question. Further, we prove that some of the graphs that we have constructed do not admit subgroups which act regularly on their vertex set; thus proving that they cannot be represented as Cayley graphs on groups.

Periodica Mathematica Hungarica, 1976

It is well-knot~n tha~ its automorphism group A(X o H) must contain the regular subgroup L G corresponding to the set of left multiplication~ by elements of G. This paper is concerned with minimizing the index [A(Xo, t/):L a] for given G, in particular when this index is always greater than 1. If G is a.beli~n but not one of seven exceptional groups, then a Cayley graph of G exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.

Proyecciones (Antofagasta), 2021

Let G be a group and S be a subset of G such that e ∉ S and S−1 ⊆ S. Then Cay(G, S) is a simple undirected Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy−1 ∈ S. The size of subset S is called the valency of Cay(G, S). In this paper, we determined the structure of all Cay(D2n, S), where D2n is a dihedral group of order 2n, n ≥ 3 and |S| = 1, 2 or 3.

Discrete Mathematics, 1995

Given a colouring A of a d-regular digraph G and a colouring H of the symmetric complete digraph on d vertices with loops, the uniformly induced colouring LnA of the line digraph LG is defined. It is shown that the group of colour-preserving automorphisms of (LG, L,A) is a subgroup of the group of colour-permuting automorphisms of (G,A). This result is then applied to prove that if (G,A) is a d-regular coloured digraph and (LG, LIIA) is a Cayley digraph, then (G, A) is itself a Cayley digraph Cay (Q, A) and H is a group of automorphisms of f2. In particular, a characterization of those Kautz digraphs which are Cayley digraphs is given. If d= 2+ for every arc-transitive digraph G, LG is a Cayley digraph when the number k of orbits by the action of the so-called Rankin group is at most 5. If k>/3 the arc-transitive k-generalized cycles for which LG is a Cayley digraph are characterized.