Cayley Digraphs Research Articles

Extremal Problems in the Construction of Distributed Loop Networks

Let $G(N, A)$ be the Cayley digraph associated with ${\bf Z}/(N)$ and $A$, where $N$ is a positive integer and $A$ is a subset of $\{1, 2, ..., N - 1\}$. Let $N(d, k)$ be the maximum $N$ such that the diameter of $G(N, A)$ is less than or equal to $d$ for some $A = \{a_{1}, a_{2},\ldots, a_{k}\}$ with $1 = a_{1}, < a_{2} < \cdots < a_{k}. An exact formula for $N(d, 2)$ is given, and $N(d, k)$ is estimated for $k \geq 3$. These results provide new bounds for minimal diameter in the construction of loop networks. A relation between this problem and the postage stamp problem in additive number theory is established to enhance the study of these problems.

Cycle prefix digraphs for symmetric interconnection networks

AbstractMotivated by the study of large graphs with given degree and diameter, and the recent interest in the design of highly symmetric interconnection networks (e.g., the study of Cayley digraphs), we are led to the search for large vertex symmetric digraphs with given degree and diameter. The main result of this paper is the construction of a new class of vertex symmetric directed graphs, Γδ(D) (δ ≥ D) that have degree δ, diameter D, and (δ + 1)δ … (δ – D + 2) vertices. The graphs Γδ(D) are first found in the notation of Cayley coset digraphs. Then, we discover that they have a very simple representation in terms of sequences like the commonly studied networks such as the hypercube, de Bruijn graphs, and Kautz graphs. Based on the sequence representation, we give a simple shortest‐path routing scheme. We also show that the average distance in our digraph Γδ(D) is very close to its diameter D. As a consequence, it follows that the natural routing scheme, which is even simpler than the shortest‐path routing, is nearly optimal on an average basis. © 1993 by John Wiley &amp; Sons, Inc.

The connectivity of hierarchical Cayley digraphs

An ordered generating set of a group is hierarchical when the group generated by the first k generators is a proper subgroup of the group generated by the first k + 1 for each k. Hierarchical Cayley graphs were introduced by Akers and Krishnamurthy in [1] and other related papers as an interesting model to build symmetrical networks. In this paper we study the connectivity of hierarchical Cayley digraphs, and we show that they have maximum connectivity except in a special case. An example of this exceptional case is given. The result generalizes similar statements of Godsil [3], Akers [1] and Hamidoune [5].

Vosperian and superconnected Abelian Cayley digraphs

A digraphX is said to be Vosperian if any fragment has cardinality either 1 or|V(X)| ? d + (X) ? 1. A digraph is said to be superconnected if every minimum cutset is the set of vertices adjacent from or to some vertex. In this paper we characterize Vosperian and superconnected Abelian Cayley directed graphs. Our main tool is a difficult theorem of J.H. Kemperman from Additive Group Theory. In particular we characterize Vosperian and superconnected loops network (also called circulants).

Vertex Transitivity and Super Line Connectedness

A connected graph X is said to be super line connected if every minimum edge cut is the set of edges incident with some vertex. It is proved that a connected Abelian Cayley graph is super line connected unless it is a cycle or the product of complete graphs, one of size 2 and one of size $m\geqq 3$. This result generalizes previous characterizations for circulants by Boesch and Wang and for hypercubes by Boesch, Lee, Wang, and Yang. A simple algebraic characterization for strongly connected Cayley digraphs that are not super arc connected and a structural characterization for the Abelian case are given. In a previous work, one of the authors presented a characterization of the connected vertex-transitive graphs that are not super line connected. The final result of the present paper generalizes this result to directed graphs.

Connectivite des graphes de cayley abeliens sans K4

We give a description for the class of connected Abelian Cayley digraphs containing no K 4 with connectivity less than the outdegree. We show that no member of this class is anti-symmetric.

Circuits in cayley digraphs of finite abelian groups

AbstractWe find all possible lengths of circuits in Cayley digraphs of two‐generated abelian groups over the two‐element generating sets and over certain three‐element generating sets.

On positive and negative atoms of Cayley digraphs

Hamidoune conjectured in [1] that a Cayley digraph always contains a positive atom; we exhibit a counterexample and thereby disprove the conjecture.

Hamiltonian paths in Cayley digraphs of finitely-generated infinite abelian groups

We show every finitely-generated, infinite abelian group (i.e. Zn x G where G is a finite abelian group) has a minimal generating set for which the Cayley digraph has a two-way infinite hamiltonian path, and if n⩾2, then this Cayley digraph also has a one-way infinite hamiltonian path. We show further that in the case of Zn (n⩾2), the Cayley digraph is 2-ply hamiltonian.

Generalized hamiltonian circuits in the cartesian product of two‐directed cycles

AbstractWe find necessary and sufficient conditions for the existence of a closed walk that traverses r vertices twice and the rest once in the Cayley digraph of Zm ⊕ Zn. This is a generalization of the results known for r = 0 or 1.

Cayley digraphs and (1, j, n)-sequencings of the alternating groups An

The problem of finding a sequencing Π 1, Π 2,… Π | A n | for the elements of the alternating group A n which minimizes the cost function C= ∑ i=1 |A n|−1 c(П i −1∘П i+1) where c( Π) = |{ j: Π( j) ≠ j}|, is solved. For n ⩾ 3 the sequencing is constructed by finding a directed Hamiltonian path in the Cayley digraph D n of A n , dtermined by the generating set B n = {(1, j, n): j ϵ { 2,3,…, n−1}}. We further consider the question of finding a directed Hamiltonian cycle in D n .

Cayley digraphs of prime-power order are hamiltonian

We find a (directed) hamiltonian circuit in any nontrivial connected Cayley digraph whose number of vertices is a prime-power.

Polynomial addition sets and polynomial digraphs

Let G be a group of order v, and f( x) be a nonzero integral polynomial. A ( v, k, f( x))-polynomial addition set in G is a subset D of G with k distinct elements such that f(Σ d∈ D d) = λΣ g∈ G g for some integer λ. We discuss some general properties of polynomial addition sets. The relation between polynomial addition sets and polynomial Cayley digraphs is studied and, in particular, some new results on Cayley x n -digraphs and strongly regular Cayley graphs are obtained. Finally, a complete list of polynomial addition sets with certain restrictions on parameters is given.

Infinite Hamiltonian paths in Cayley digraphs

Let Cay( S: H) be the Cayley digraph of the generators S in the group H. A one-way infinite Hamiltonian path in the digraph G is a listing of all the vertices [ ν i :1⩽ i<∞], such that there is an arc from ν i to ν 1+1. A two-way infinite Hamiltonian path is similarly defined, with i ranging from −∞ to ∞. In this paper, we give conditions on S and H for the existence of one- and two-way infinite Hamiltonian paths in Cay( X: H). Two of our results can be summarized as follows. First, if S is countably infinite and H is abelian, then Cay( S: H) has one- and two-way Hamiltonian paths if and only if it is strongly connected (except for one infinite family). We also give necessary and sufficient conditions on S for Cay( S: H) to be strongly connected for a large class of Cayley digraphs. Second, we show that any Cayley digraph of a countable locally finite group has both one- and two-way infinite Hamiltonian paths. As a lemma, we give a relation between the strong connectivity and the outer valence of finite vertex-transitive digraphs.

Sur la separation dans les graphes de Cayley abeliens

In this note, we prove that Cayley digraphs on Abelian groups have some separation properties similar to undirected graphs. L'objet de cette note est de montrer que les graphes orientés de Cayley définis sur les groupes abéliens ont des propriétés de séparation similaires à celles des graphes non orientés.

On the Connectivity of Cayley Digraphs

We prove that the atom of a Cayley diagraph which contains the unity is a subgroup. As an application we obtain a short proof for a theorem of Imrich on the connectivity of the assignment polytope. We prove also that the connectivity of a Cayley digraph defined by a minimal generating set is equal to its indegree. This result generalizes a theorem of Godsil in the undirected case. We construct a class of Cayley digraphs with optimal connectivity. A symmetric subclass of our class was constructed by Boesch and Felger.

On hamiltonian circuits in cartesian products of Cayley digraphs

In this paper, we investigate the existence of a hamiltonian circuit in the cartesian product of two Cayley digraphs. Three of our results can be summarized as follows. Suppose K is the Cayley digraph of a dihedral, semidihedral, or dicyclic group arising from a specified pair of (standard) generators, and suppose L is a Cayley digraph with a hamiltonian circuit. Then, the cartesian product of K and L has a hamiltonian circuit. As a corollary to our main theorem, we also show that the cartesian product of an undirected cycle of length n and a directed cycle of length k has a hamiltonian circuit unless n = 2 and k is odd. Some open problems are stated.

The conjunction of Cayley digraphs

If G1 and G2 are digraphs with vertex sets V1 and V2, then the conjunction of G1 and G2, denoted G1 · G2, is a digraph with vertex set V1 × V2 in which there is an arc from (x1, x2) to (y1, y2) if there is an arc from x1 to y1 in G1 and an arc from x2 to y2 in G2. If Cay(S) and Cay(T) are Cayley digraphs, then Cay(S)·Cay(T) = Cay(S×T) as long as the digraph on the left is connected. In this paper we find sufficient conditions for the conjunction of two Cayley digraphs to have a hamiltonian circuit. For certain classes of generating sets S and T, we characterize the cases for which Cay(S) · Cay(T) has a hamiltonian circuit. We also present some basic results on the existence of hamiltonian paths in the conjunction of two Cayley digraphs.