Regular sets in Cayley graphs

Characterization of subgroup perfect codes in Cayley graphs

Nearly perfect sets in graphs

The induced subgraph of a unit graph with vertex set as the idempotent elements of a ring R is a graph which is obtained by deleting all non idempotent elements of R. Let C be a subset of the vertex set in a graph Γ. Then C is called a perfect code if for any x, y ∈ C the union of the closed neighbourhoods of x and y gives the the vertex set and the intersection of the closed neighbourhoods of x and y gives the empty set. In this paper, the perfect codes in induced subgraphs of the unit graphs associated with the ring of integer modulo n, Zn that has the vertex set as idempotent elements of Zn are determined. The rings of integer modulo n are classified according to their induced subgraphs of the unit graphs that accept a subset of a ring Zn of different sizes as the perfect codes

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.

1·1. We shall write for the class of groups all of whose non-trivial normal subgroups have finite index. Thus, rather obviously, finite groups, simple groups, and the infinite cyclic and dihedral groups all lie in the class . Other examples of -groups are to be found in a variety of contexts. The main result of Mennicke (8) shows that, for n ≥ 3, the factor group of SL(n, Z) by its centre is a -group, where SL(n, Z) denotes the unimodular group of all n × n invertible matrices with integer coefficients and with determinant 1. In McLain(7), an example is given of an infinite, locally finite, locally soluble -group, whose only non-trivial normal subgroups are the terms of its derived series, and in (4), P. Hall discusses an infinite -group, all of whose proper quotients are finite p-groups. If is a class of groups closed under taking homomorphic images, it is easily seen that the existence of an infinite -group satisfying the maximal condition for normal subgroups implies the existence of an infinite -group in the class , so that certain questions concerning the finiteness of groups satisfying the maximal condition for normal subgroups can be interpreted as questions about -groups.

The search for perfect error-correcting codes has received intense interest since the seminal work by Hamming. Decades ago, Golomb and Welch studied perfect codes for the Lee metric in multidimensional torus constellations. In this work, we focus our attention on a new class of four-dimensional signal spaces which include tori as subcases. Our constellations are modeled by means of Cayley graphs defined over quotient rings of Lipschitz integers. Previously unexplored perfect codes of length one will be provided in a constructive way by solving a typical problem of vertices domination in graph theory. The codewords of such perfect codes are constituted by the elements of a principal (left) ideal of the considered quotient ring. The generalization of these techniques for higher dimensional spaces is also considered in this work by modeling their signal sets through Cayley-Dickson algebras.

Let [Formula: see text] be a graph. A subset [Formula: see text] of [Formula: see text] is called a perfect code of [Formula: see text], when [Formula: see text] is an independent set and every vertex of [Formula: see text] is adjacent to exactly one vertex in [Formula: see text]. Let [Formula: see text] = Cay(G, S) be a Cayley graph of a finite group [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is called a perfect code of [Formula: see text], when there exists a Cayley graph [Formula: see text] of [Formula: see text] such that [Formula: see text] is a perfect code of [Formula: see text]. Recently, groups whose set of all subgroup perfect codes forms a chain are classified. Also, groups with no proper nontrivial subgroup perfect code are characterized. In this paper, we generalize it and classify groups whose set of all non-perfect code subgroups forms a chain.

Symmetric graphs of valency five and their basic normal quotients

A Cayley snark is a cubic Cayley graph which is not 3-edge-colourable. In the paper we discuss the problem of the existence of Cayley snarks. This problem is closely related to the problem of the existence of non-hamiltonian Cayley graphs and to the question whether every Cayley graph admits a nowhere-zero 4-flow. So far, no Cayley snarks have been found. On the other hand, we prove that the smallest example of a Cayley snark, if it exists, comes either from a non-abelian simple group or from a group which has a single non-trivial proper normal subgroup. The subgroup must have index two and must be either non-abelian simple or the direct product of two isomorphic non-abelian simple groups.

In answer to a question of P. Hall, we supply another construction of a group which is isomorphic to each of its non-trivial normal subgroups.

On generalized perfect difference sets constructed from Sidon sets

A perfect code in a graph $$\Gamma $$ is a subset C of $$V(\Gamma )$$ such that no two vertices in C are adjacent and every vertex in $$V(\Gamma ){\setminus } C$$ is adjacent to exactly one vertex in C. Let G be a finite group and C a subset of G. Then C is said to be a perfect code of G if there exists a Cayley graph of G admiting C as a perfect code. It is proved that a subgroup H of G is a perfect code of G if and only if a Sylow 2-subgroup of H is a perfect code of G. This result provides a way to simplify the study of subgroup perfect codes of general groups to the study of subgroup perfect codes of 2-groups. As an application, a criterion for determining subgroup perfect codes of projective special linear groups $$\textrm{PSL}(2,q)$$ is given.

Efficient dominating sets in circulant graphs

A perfect code in a graph $$\Gamma = (V, E)$$ Γ = ( V , E ) is a subset C of V such that no two vertices in C are adjacent and every vertex in $$V \setminus C$$ V \ C is adjacent to exactly one vertex in C . A total perfect code in $$\Gamma $$ Γ is a subset C of V such that every vertex of $$\Gamma $$ Γ is adjacent to exactly one vertex in C . In this paper we prove several results on perfect codes and total perfect codes in Cayley graphs of finite abelian groups.

The study of graph properties has gathered many attentions in the past years. The graph properties that are commonly studied include the chromatic number, the clique number and the domination number of a finite graph. In this study, a type of graph properties, which is the perfect code is studied. The perfect code is originally used in coding theory, then extended to other fields including graph theory. Hence, in this paper, the perfect code is determined for the commuting zero divisor graphs of some finite rings of matrices. First, the commuting zero divisor graph of the finite rings of matrices is constructed where its vertices are all zero divisors of the ring and two distinct vertices, say x and y, are adjacent if and only if xy = yx = 0. Then, from the vertex set of the graph, the neighborhood elements of the vertices are determined in order to compute the perfect codes of the graph.