Cayley Digraphs Research Articles

Weak Hopf Algebra and Its Quiver Representation

This study induced a weak Hopf algebra from the path coalgebra of a weak Hopf quiver. Moreover, it gave a quiver representation of the said algebra which gives rise to the various structures of the so-called weak Hopf algebra through the quiver. Furthermore, it also showed the canonical representation for each weak Hopf quiver. It was further observed that a Cayley digraph of a Clifford monoid can be embedded in its corresponding weak Hopf quiver of a Clifford monoid. This lead to the development of the foundation structures of weak Hopf algebra. Such quiver representation is useful for the classification of its path coalgebra. Additionally, some structures of module theory of algebra were also given. Such algebras can also be applied for obtaining the solutions of “quantum Yang–Baxter equation” that has many applications in the dynamical systems for finding interesting results.

On the asymptotic enumeration of Cayley graphs

In this paper, we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group R has a nontrivial proper normal subgroup N with the property that the automorphism group of the digraph fixes each N-coset setwise, and the cases where it does not. In this paper, we deal with undirected graphs in the case where the regular group has such a nontrivial proper normal subgroup.

Domination parameters on Cayley digraphs of transformation semigroups with fixed sets

For a nonempty subset $Y$ of a nonempty set $X$, denote by $Fix(X,Y)$ the semigroup of full transformations on the set $X$ in which all elements in $Y$ are fixed. The Cayley digraph $Cay$ $(Fix(X,Y),A)$ of $Fix(X,Y)$ with respect to a connection set $A\subseteq Fix(X,Y)$ is defined as a digraph whose vertex set is $Fix(X,Y)$ and two vertices $\alpha, \beta$ are adjacent in sense of drawing a directed edge (arc) from $\alpha$ to $\beta$ if there exists $\mu\in A$ such that $\beta = \alpha\mu$. In this paper, we determine domination parameters of $Cay$ $(Fix(X,Y),A)$ where $A$ is a subset of $Fix(X,Y)$ related to minimal idempotents and permutations in $Fix(X,Y)$

Asymptotic enumeration of Cayley digraphs

In this paper we show that almost all Cayley digraphs have automorphism group as small as possible; that is, they are digraphical regular representations (DRRs). More precisely, we show that as r tends to infinity, for every finite group R of order r, out of all possible Cayley digraphs on R the proportion whose automorphism group is as small as possible tends to 1. This proves a natural conjecture first proposed in 1982 by Babai and Godsil.

A Note on Moore Cayley Digraphs

Let $$\varDelta$$ be a digraph of diameter 2 with the maximum undirected vertex degree t and the maximum directed out-degree z. The largest possible number v of vertices of $$\varDelta$$ is given by the following generalization of the Moore bound: $$\begin{aligned} v\le (z+t)^2+z+1, \end{aligned}$$ and a digraph attaining this bound is called a Moore digraph. Apart from the case $$t=1$$ , only three Moore digraphs are known, which are also Cayley graphs. Using computer search, Erskine (J Interconnect Netw, 17: 1741010, 2017) ruled out the existence of further examples of Cayley digraphs attaining the Moore bound for all orders up to 485. We use an algebraic approach to this problem, which goes back to an idea of G. Higman from the theory of association schemes, also known as Benson’s Lemma in finite geometry, and show non-existence of Moore Cayley digraphs of certain orders.

Arc and twin arc domination in Cayley digraphs

Let G be a finite group and let S be a generating subset of G. The Cayley digraph Cay (G, S) is the digraph with node set as the elements of G and there is an arc from u to us whenever u ϵ G and s ϵ S. An arc e1 in a digraph D out arc dominates itself as well as all arcs ei. Such that {e1, ei} is a directed path of length 2 in D. While e1 in arc dominates both itself and all arcs ej such that {ei, e1} is a directed path of length 2 in D. The arc domination number is the minimum cardinality of an out arc dominating set of D denoted by γ′(D). A set of edges of D is twin arc dominating set if every edge of D is out arc dominated by some edge of S and in arc dominated by some edge of S. The minimum Cardinality of a twin arc dominating set is the twin arc domination number denoted by γ* (D) of D. This paper discusses the arc domination and twin arc domination of Cayley digraphs and attempts to find bounds for the arc domination number of the same class of graphs.

Cayley graphs of more than one abelian group

We show that for certain integers n, the problem of whether or not a Cayley digraph Γ of ℤn is also isomorphic to a Cayley digraph of some other abelian group G of order n reduces to the question of whether or not a natural subgroup of the full automorphism group contains more than one regular abelian group up to isomorphism (as opposed to the full automorphism group). A necessary and sufficient condition is then given for such circulants to be isomorphic to Cayley digraphs of more than one abelian group, and an easy-to-check necessary condition is provided.

The Cayley isomorphism property for the group

A finite group G is called a -group if every two isomorphic Cayley digraphs over G are Cayley isomorphic, i.e. their connection sets are conjugate by a group automorphism. We prove that the group where p is a prime, is a -group if and only if

The Cayley isomorphism property for the group C^5_2 × C_p

A finite group $G$ is called a DCI-group if two Cayley digraphs over $G$ are isomorphic if and only if their connection sets are conjugate by a group automorphism. We prove that the group $C_2^5\times C_p$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$. Together with the previously obtained results, this implies that a group $G$ of order $32p$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$ and $G\cong C_2^5\times C_p$.

Recognizing Cayley Digraphs

SummaryA criterion is developed that decides in a finite number of steps whether a finite digraph is a Cayley digraph of some group. In the process, a class of algebraic structure more general than groups are considered that arise naturally from digraphs that are “almost” Cayley.

A method to determine algebraically integral Cayley digraphs on finite abelian group

A method to determine algebraically integral Cayley digraphs on finite abelian group

On strongly sequenceable abelian groups

A group is strongly sequenceable if every connected Cayley digraph on the group admits an orthogonal directed cycle or an orthogonal directed path. This paper deals with the problem of whether finite abelian groups are strongly sequenceable. A method based on posets is used to show that if the connection set for a Cayley digraph on an abelian group has cardinality at most nine, then the digraph admits either an orthogonal directed path or an orthogonal directed cycle.

ON CONNECTEDNESS AND COMPLETENESS OF CAYLEY DIGRAPHS OF TRANSFORMATION SEMIGROUPS WITH FIXED SETS

Let $\text{Fix}(X,Y)$ be a semigroup of full transformations on a set $X$ in which elements in a nonempty subset $Y$ of $X$ are fixed. In this paper, we construct the Cayley digraphs of $\text{Fix}(X,Y)$ and study some structural properties of such digraphs such as the connectedness and the completeness. Further, some prominent results of Cayley digraphs of $\text{Fix}(X,Y)$ relative to minimal idempotents are verified. In addition, the characterization of an equivalence digraph of the Cayley digraph of $\text{Fix}(X,Y)$ is also investigated.

On the existence and the enumeration of bipartite regular representations of Cayley graphs over abelian groups

AbstractIn this paper, we are interested in the asymptotic enumeration of bipartite Cayley digraphs and Cayley graphs over abelian groups. Let be an abelian group and let be the automorphism of defined by , for every . A Cayley graph is said to have an automorphism group as small as possible if . In this paper, we show that, except for two infinite families, almost all bipartite Cayley graphs on abelian groups have automorphism group as small as possible. We also investigate the analogous question for bipartite Cayley digraphs. These results are used for the asymptotic enumeration of bipartite Cayley digraphs and graphs over abelian groups.

Hermitian Adjacency Spectrum of Cayley Digraphs over Dihedral Group

We first study the spectrum of Hermitian adjacency matrix (H-spectrum) of Cayley digraphs X(D2n, S) on dihedral group D2n with |S| = 3. Then we show that all Cayley digraphs X(D2p, S) with |S| = 3 and p odd prime are Cay-DS, namely, for any Cayley digraph X(D2p, T), X(D2p, T) and X(D2p, S) having the same H-spectrum implies that they are isomorphic.

On the automorphism groups of vertex-transitive Cayley digraphs of monoids

In 1982, Babai and Godsil conjectured that almost all Cayley digraphs are digraphical regular representations. In 1998, Xu conjectured that almost all Cayley digraphs are normal [i.e., \(G_L\) is a normal subgroup of the automorphism group of \(\text {Cay}(G,C)\)]. Finally, in 1994, Praeger and Mckay conjectured that almost all undirected vertex-transitive graphs are Cayley graphs of groups. In this paper, first we present the variants of these conjectures for Cayley digraphs of monoids and we determine when the variant of Babai and Godsil’s conjecture is equivalent to the variant of Xu’s conjecture. Then, as a special consequence of our results, we conclude that Xu’s conjecture is equivalent to Babai and Godsil’s conjecture. On the other hand, we give affirmative answer to the variant of Praeger and Mckay’s conjecture and we prove that a Cayley digraph of a monoid is vertex-transitive if and only if it is isomorphic to a Cayley digraph of a group. Finally, we use this characterization to give an affirmative answer to a question raised by Kelarev and Praeger about vertex-transitivity of Cayley digraphs of monoids. Also using this characterization, we explicitly determine the automorphism groups of vertex-transitive Cayley digraphs of monoids.

Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups

Let Γ = Cay(G, S) be a Cayley digraph on a group G and let A = Aut(Γ). The Cayley index of Γ is |A : G|. It has previously been shown that, if p is a prime, G is a cyclic p-group and A contains a noncyclic regular subgroup, then the Cayley index of Γ is superexponential in p. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if p is an odd prime and G is abelian but not cyclic, and has order a power of p at least p3, then there is a Cayley digraph Γ on G whose Cayley index is just p, and whose automorphism group contains a nonabelian regular subgroup.

Divisible design Cayley digraphs

Divisible design digraphs which can be obtained as Cayley digraphs are studied. A characterization of divisible design Cayley digraphs in terms of the generating sets is given. Further, we give several constructions of divisible design Cayley digraphs and classify divisible design Cayley digraphs on v≤27 vertices.

Metric dimension of Cayley digraphs of split metacyclic groups

A directed Cayley graph Cay(Γ,X) is specified by a group Γ and an identity-free generating set X for this group. Vertices of Cay(Γ,X) are elements of Γ and there is a directed edge from a vertex u to a vertex v in Cay(Γ,X) if and only if there is a generator x∈X such that ux=v. We study the metric dimension for the directed Cayley graphs Cay(Γs,{a,b}) of general split metacyclic groups, and present the exact values of the metric dimension for the special split metacyclic groups Γs=〈a,b|an=b2s=1,ba=a−1b〉.

Domination of Cayley Digraph and its Complement

A Cayley graph constructed out of a group Γ and its generating set A is denoted by Cay (Γ, A). The digraph with the same node set as the original digraph is said to be a complement digraph if it has an edge from x to y exactly when the original digraph does not have an edge from x to y. A subset Ɖ of V is called a dominating set if each vertex in V- Ɖ is adjacent to at least one vertex in Ɖ. The minimum cardinality of a dominating set is called Domination number which is denoted by γ. The domination number of Cayley digraphs and Complement of Cayley digraphs of groups are investigated in this paper. Also, the graph relationship involving domination parameters in a graph and its complement are studied.