Cayley Digraphs Research Articles

Perfect codes in 2-valent Cayley digraphs on abelian groups

For a digraph Γ, a subset C of V(Γ) is a perfect code if C is a dominating set such that every vertex of Γ is dominated by exactly one vertex in C. In this paper, we classify strongly connected 2-valent Cayley digraphs on abelian groups admitting a perfect code, and determine completely all perfect codes of such digraphs.

On oriented m $m$‐semiregular representations of finite groups

AbstractA finite group admits an oriented regular representation if there exists a Cayley digraph of such that it has no digons and its automorphism group is isomorphic to . Let be a positive integer. In this paper, we extend the notion of oriented regular representations to oriented ‐semiregular representations using ‐Cayley digraphs. Given a finite group , an ‐Cayley digraph of is a digraph that has a group of automorphisms isomorphic to acting semiregularly on the vertex set with orbits. We say that a finite group admits an oriented ‐semiregular representation (OSR for short) if there exists an ‐Cayley digraph of such that it has no digons and is isomorphic to its automorphism group. Moreover, if is regular, that is, each vertex has the same in‐ and out‐valency, we say is a regular oriented ‐semiregular representation (regular OSR for short) of . In this paper, we classify finite groups admitting a regular OSR or an OSR for each positive integer .

On the path homology of Cayley digraphs and covering digraphs

We develop a theory of covering digraphs, similar to the theory of covering spaces. By applying this theory to Cayley digraphs, we build a “bridge” between GLMY-theory and group homology theory, which helps to reduce path homology calculations to group homology computations. We show some cases where this approach allows us to fully express path homology in terms of group homology. To illustrate this method, we provide a path homology computation for the Cayley digraph of the additive group of rational numbers with a generating set consisting of inverses to factorials. The main tool in our work is a filtered simplicial set associated with a digraph, which we call the filtered nerve of a digraph.

Perfect directed codes in Cayley digraphs

<p>A perfect directed code (or an efficient twin domination) of a digraph is a vertex subset where every other vertex in the digraph has a unique in- and a unique out-neighbor in the subset. In this paper, we show that a digraph covers a complete digraph if and only if the vertex set of this digraph can be partitioned into perfect directed codes. Equivalent conditions for subsets in Cayley digraphs to be perfect directed codes are given. Especially, equivalent conditions for normal subsets, normal subgroups, and subgroups to be perfect directed codes in Cayley digraphs are given. Moreover, we show that every subgroup of a finite group is a perfect directed code for a transversal Cayley digraph.</p>

The Vertex Distance on Cayley Digraphs of Rectangular Groups with respect to the Cartesian Product Connection Sets

A rectangular group S is a semigroup isomorphic to the direct product of a group, a left zero semigroup, and a right zero semigroup. The Cayley digraph Cay(S,A) of a finite rectangular group S with respect to a nonempty subset A of S is defined as a digraph with vertex set S and arc set consisting of ordered pairs (u,ua) in SxS for some a in A. The set A is called a connection set of Cay(S,A). Let u and v be in S. The distance from u to v in Cay(S,A), denoted by d(u,v), is the number of arcs in the shortest directed u-v path if one exists and is infinity otherwise. In this paper, we characterize the distances of any two vertices in Cayley digraphs of rectangular groups with respect to the Cartesian product connection sets

Fractional domination and fractional total domination on Cayley digraphs of transformation semigroups with fixed sets

<abstract><p>For a set $ X $ and a nonempty subset $ Y $ of $ X $, denote by $ T(X) $ the full transformation semigroup under the composition whose elements are functions on $ X $. Let $ Fix(X, Y) $ be the subsemigroup of $ T(X) $ containing functions $ \alpha\in T(X) $ in which each element in $ Y $ is a fixed point of $ \alpha $. Moreover, let $ A $ be a nonempty subset of $ Fix(X, Y) $. The Cayley digraph of $ Fix(X, Y) $ with respect to a connection set $ A $ is a digraph with vertex set $ Fix(X, Y) $ and two vertices $ \alpha, \beta $ induce an arc $ (\alpha, \beta) $ if $ \beta = \alpha\lambda $ for some $ \lambda\in A $. In this paper, the concepts of fractional dominating and fractional total dominating functions of those Cayley digraphs were investigated. Furthermore, the fractional domination and fractional total domination numbers were determined.</p></abstract>

Fault-Tolerant Metric Dimension of Barycentric Subdivision of Cayley Graphs

Metric dimension and fault-tolerant metric dimension of any graph G is subject to size of resolving set. It has become more important in modern GPS and sensors based world as resolving set ensures that in case of semi outage system is still scalable using redundant interfaces. Metric dimension of several interesting classes of graphs have been investigated like Cayley digraphs, Cartesian product of graphs, wheel graphs, convex polytopes and certain networks for categorical product of graphs. In this paper we used the phenomena of barycentric subdivision of graph and proved that fault-tolerant metric dimension of barycentric subdivision of Cayley graph is constant.

Isoperimetric stability in lattices

We obtain isoperimetric stability theorems for general Cayley digraphs on Z d {\mathbb {Z}}^d . For any fixed B B that generates Z d {\mathbb {Z}}^d over Z {\mathbb {Z}} , we characterise the approximate structure of large sets A A that are approximately isoperimetric in the Cayley digraph of B B : we show that A A must be close to a set of the form k Z ∩ Z d kZ \cap {\mathbb {Z}}^d , where for the vertex boundary Z Z is the conical hull of B B , and for the edge boundary Z Z is the zonotope generated by B B .

Algebraic degrees of 2-Cayley digraphs over abelian groups

A digraph Γ is called a 2-Cayley digraph over a group G if there exists a 2-orbit semiregular subgroup of Aut(Γ) isomorphic to G. In this paper, we give upper and lower bounds for the algebraic degrees of 2-Cayley digraphs over abelian groups. As an application, we consider the Cayley digraphs over finite groups admitting an abelian subgroup of index 2. Special attention is paid for Cayley graphs over generalized dihedral groups and generalized dicyclic groups. For Cayley graphs over generalized dihedral groups and generalized dicyclic groups, we completely determine their algebraic degrees and splitting fields. At last, the upper and lower bounds for the algebraic degrees of Cayley digraphs over semi-dihedral groups are obtained.

On the Independence Number of Cayley Digraphs of Clifford Semigroups

Let S be a Clifford semigroup and A a subset of S. We write Cay(S,A) for the Cayley digraph of a Clifford semigroup S relative to A. The (weak, path, weak path) independence number of a graph is the maximum cardinality of an (weakly, path, weakly path) independent set of vertices in the graph. In this paper, we characterize maximal connected subdigraphs of Cay(S,A) and apply these results to determine the (weak, path, weak path) independence number of Cay(S,A).

Normal Cayley digraphs of dihedral groups with CI-property

A Cayley (di)graph Cay(G,S) of a group G with respect to a set S ⊆ G is said to be normal if the image of G under its right regular representation is normal in the automorphism group of Cay(G,S), and is called a CI-(di)graph if for every T ⊆ G with Cay(G,S)≅ Cay(G,T), there is α∈ Aut(G) such that Sα = T. A finite group G is called a DCI-group or an NDCI-group if all Cayley digraphs or all normal Cayley digraphs of G are CI-digraphs, respectively, and is called a CI-group or an NCI-group if all Cayley graphs or all normal Cayley graphs of G are CI-graphs, respectively. Motivated by a conjecture proposed by Ádám in 1967, CI-groups and DCI-groups have been actively studied during the last fifty years by many researchers in algebraic graph theory. It took about thirty years to obtain the classification of cyclic CI-groups and DCI-groups, and recently, the first two authors, among others, classified cyclic NCI-groups and NDCI-groups. Even though there are many partial results on dihedral CI-groups and DCI-groups, their classification is still elusive. In this paper, we prove that a dihedral group of order 2n is an NCI-group or an NDCI-group if and only if n = 2, 4 or n is odd. As a direct consequence, we have that if a dihedral group D2n of order 2n is a DCI-group then n = 2 or n is odd-square-free, and that if D2n is a CI-group then n = 2, 9 or n is odd-square-free, throwing some new light on classification of dihedral CI-groups and DCI-groups. As a byproduct, we construct a non-CI Cayley graph of the dihedral group D8, but Holt and Royle [A census of small transitive groups and vertex-transitive graphs, J. Symbolic Comput. 101 (2020) 51-60] claimed that D8 is a CI-group by an algorithm there.

Moore mixed graphs from Cayley graphs

A Moore ( r , z , k ) -mixed graph G has every vertex with undirected degree r , directed in- and out-degree z , diameter k , and number of vertices (or order) attaining the corresponding Moore bound M ( r , z , k ) for mixed graphs. When the order of G is close to M ( r , z , k ) vertices, we refer to it as an almost Moore graph. The first part of this paper is a survey about known Moore (and almost Moore) general mixed graphs that turn out to be Cayley graphs. Then, in the second part of the paper, we give new results on the bipartite case. First, we show that Moore bipartite mixed graphs with diameter three are distance-regular, and their spectra are fully characterized. In particular, an infinity family of Moore bipartite (1, z , 3) -mixed graphs is presented, which are Cayley graphs of semidirect products of groups. Our study is based on the line digraph technique, and on some results about when the line digraph of a Cayley digraph is again a Cayley digraph.

Structural properties and isomorphism theorems for Cayley digraphs of full transformation semigroups with respect to Green's equivalence classes

Let T(X) be the full transformation semigroup on a nonempty set X. In this paper, the Cayley digraphs of T(X) with connection sets L and R, the Green's equivalence classes of T(X) according to the Green's relations L and R, are investigated. Furthermore, their connectedness properties are characterized. In addition, the isomorphism theorems for Cayley digraphs of T(X) are also presented.

An analogue of Furstenberg–Sárközy’s theorem and an alternative solution to Waring’s problem over finite fields

In this paper, we use Cayley digraphs to obtain some new self-contained proofs for Waring’s problem over finite fields, proving that any element of a finite field Fq can be written as a sum of m many kth powers as long as q>k2mm−1; and we also compute the smallest positive integers m such that every element of Fq can be written as a sum of m many kth powers for all q too small to be covered by the above mentioned results when 2⩽k⩽37.In the process of developing the proofs mentioned above, we arrive at another result (providing a finite field analogue of Furstenberg–Sárközy’s Theorem) showing that any subset E of a finite field Fq for which |E|>qkq−1 must contain at least two distinct elements whose difference is a kth power.

Eigenvalues of Cayley Graphs

We survey some of the known results on eigenvalues of Cayley graphs and their applications, together with related results on eigenvalues of Cayley digraphs and generalizations of Cayley graphs.

Groups for which it is easy to detect graphical regular representations

We say that a finite group G is DRR-detecting if, for every subset S of G, either the Cayley digraph Cay(G,S) is a digraphical regular representation (that is, its automorphism group acts regularly on its vertex set) or there is a nontrivial group automorphism φ of G such that φ(S) = S. We show that every nilpotent DRR-detecting group is a p-group, but that the wreath product Zp wr Zp is not DRR-detecting, for every odd prime p. We also show that if G1 and G2 are nontrivial groups that admit a digraphical regular representation and either gcd(|G1|, |G2|) = 1, or G2 is not DRR-detecting, then the direct product G1 x G2 is not DRR-detecting. Some of these results also have analogues for graphical regular representations.

Normal Cayley digraphs of generalized quaternion groups with CI-property

A Cayley digraph Cay(G,S) of a finite group G with respect to a subset S of G, where S does not contain the identity 1 of G, is said to be a CI-digraph, if Cay(G,S)≅Cay(G,T) implies that G has an automorphism mapping S to T. The group G is called a DCI-group or an NDCI-group if all Cayley digraphs or normal Cayley digraphs of G are CI-digraphs. We prove in this paper that a generalized quaternion group Q4n of order 4n is an NDCI-group if and only if n=2 or n is odd. As a result, we show that if Q4n is a DCI-group then n=2 or n is odd-square-free.

Structural Properties and Isomorphism Theorems for Cayley Digraphs of Full Transformation Semigroups with Respect to Green's Equivalence Classes

Structural Properties and Isomorphism Theorems for Cayley Digraphs of Full Transformation Semigroups with Respect to Green's Equivalence Classes

Normal Cayley digraphs of cyclic groups with CI-property

A Cayley (di)graph of a group G is called normal if the right regular representation of G is normal in the full automorphism group of and a CI-(di)graph if for every Cayley (di)graph implies that there is such that We call a group G an NDCI-group or NCI-group if all normal Cayley digraphs or graphs of G are CI-digraphs or CI-graphs, respectively. We prove that a cyclic group of order n is an NDCI-group if and only if and an NCI-group if and only if either n = 8 or

The group [formula omitted] is a DCI-group

We prove that the group Cp4×Cq is a DCI-group for distinct primes p and q, that is, two Cayley digraphs over Cp4×Cq are isomorphic if and only if their connection sets are conjugate by a group automorphism.