Circulant Digraphs Research Articles

A Classification of DCI(CI)-Subsets for Cyclic Group of Odd Prime Power Order

Let Zn be a cyclic group of order n and C(Zn, S) the circulant digraph of Zn with respect to S⊆Zn\{o}. S is called a DCI-subset of Zn if, for any circulant digraph C(Zn, T), C(Zn, S)≅C(Zn, T) implies that S and T are conjugate in Aut(Zn), the automorphism group of Zn. In this paper, we give a complete classification for DCI-subsets of Zn in the case of n=pa, where p is an odd prime.

On non‐Hamiltonian circulant digraphs of outdegree three

We construct infinitely many connected, circulant digraphs of outdegree three that have no Hamiltonian circuit. All of our examples have an even number of vertices, and our examples are of two types: either every vertex in the digraph is adjacent to two diametrically opposite vertices, or every vertex is adjacent to the vertex diametrically opposite to itself. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 319–331, 1999

On non-Hamiltonian circulant digraphs of outdegree three

We construct infinitely many connected, circulant digraphs of outdegree three that have no Hamiltonian circuit. All of our examples have an even number of vertices, and our examples are of two types: either every vertex in the digraph is adjacent to two diametrically opposite vertices, or every vertex is adjacent to the vertex diametrically opposite to itself. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 319–331, 1999

Asymptotic enumeration theorems for the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs

The asymptotic properties of the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs are studied. Let\(C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)\) be a directed circulant graph. Let\(\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)\) and\(\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)\) be the numbers of spanning trees and of Eulerian trails, respectively. Then $$\begin{array}{*{20}c} \begin{gathered} \lim \frac{1}{k}\sqrt[p]{{T\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)}} = 1, \hfill \\ \lim \frac{1}{{k!}}\sqrt[p]{{E\left( {C\left( {p,s_1 ,s_2 , \cdots ,s_k } \right)} \right)}} = 1, \hfill \\ \end{gathered} & {p \to \infty .} \\ \end{array} $$ Furthermore, their line digraph and iterations are dealt with and similar results are obtained for undirected circulant graphs.

On arc-transitive circulant digraphs

Denote by C,(S) the circulant digraph with vertex set Z,, = {0,1,2,. n−1} and symbol set S(≠-S)⊆∖{0}. Let X be the automorphism group of C n (.S) and X,, the stabilizer of 0 in X. ThenCn(S) is arc-transitive if and only if Xoacts transitively onS. In this paper, Cn(S) with Xo|s being the symmetric group is characterized by its symbol set. By the way all the arctransitive circulant digraphs of degree 2 and 3 are given.

Hamiltonian cycles in circulant digraphs with two stripes

The circulant traveling salesman problem ( CTSP) is the problem of finding a minimum weight Hamiltonian cycle in a weighted graph with circulant distance matrix. The computational complexity of this problem is not known. In fact, even the complexity of deciding Hamiltonicity of the underlying graph is unknown. This paper provides a characterization of Hamiltonian digraphs with circulant distance matrix containing only two nonzero stripes. The corresponding conditions can be checked in polynomial time. Secondly, we show that all Hamiltonian cycles of a circulant 2-digraph are periodic. Based on these two results, a method for enumerating all Hamiltonian cycles in such digraphs is described. Moreover, two simple algorithms are derived for solving the sum and bottleneck versions of CTSP for circulant distance matrices with two nonzero stripes.

The Automorphism Groups of Minimal Infinite Circulant Digraphs

An infinite circulant digraph is a Cayley digraph of the cyclic group ofZof integers. Here we prove that the full automorphism group of any strongly connected infinite circulant digraph over minimal generating set is just the group of translations ofZ.We also present some related conjectures.

Characterization of c-circulant digraphs of degree two which are circulant

A c-circulant digraph G N ( c, Δ) has Z N as its vertex set and adjacency rules given by x → cx + a with α ϵ Δ ⋐ Z N . The c-circulant digraphs of degree two which are isomorphic to some circulant digraph are characterized, and the corresponding isomorphism is given. Moreover, a sufficient condition is obtained for a c-circulant digraph to be a Cayley digraph.

On circulant digraphs with regular automorphism groups

We prove that the cyclic groupZ n (n ≥ 3) has ak-regular digraph regular representation if and only if 0 <k <n − 1.

Isomorphisms of circulant digraphs of degree 3

In this paper, we introduce a new approach to characterize the isomorphisms of circulant digraphs. In terms of this method, we completely determine the isomorphic classes of circulant digraphs of degree 3. In particular, we characterize those circulant digraphs of degree 3 which don’t satisfy Adam’s conjecture.

Connectivities of random circulant digraphs

In this paper, we prove that almost all circulant digraphs are strongly connected. Furthermore, for any given positive integerm, we show that almost every circulant digraphC has connectivity at least\(\frac{m}{{1 + m}} + \left( {d\left( C \right) + 1} \right)\), whered(C) is the vertex degree ofC.

Double commutative-step digraphs with minimum diameters

From a natural generalization to Z 2 of the concept of congruence, it is possible to define a family of 2-regular digraphs that we call double commutative-step digraphs. They turn out to be Cayley diagrams of Abelian groups generated by two elements, and can be represented by L-shaped tiles which tessellate the plane periodically. A double commutative-step digraph with n vertices is said to be tight if its diameter attains the lower bound lb(n)= 3n ˥−2 . In this paper, the tiles associated with tight double commutative-step digraphs are fully characterized. This allows us to construct such digraphs and also to find for which values of n they exist. In particular, we characterize a complete set of families of tight double fixed-step digraphs (also called circulant digraphs), generalizing some previously known results.

Connectivity of circulant digraphs

AbstractAn explicit expression is derived for the connectivity of circulant digraphs.

A class of vertex-transitive digraphs, II

(1). We determine the number of non-isomorphic classes of self-complementary circulant digraphs with pq vertices, where p and q are distinct primes. The non-isomorphic classes of these circulant digraphs with pq vertices are enumerated. (2). We also determine the number of non-isomorphic classes of self-complementary, vertex-transitive digraphs with a prime number p vertices, and the number of self-complementary strongly vertex-transitive digraphs with p vertices. The non-isomorphic classes of strongly vertex-transitive digraphs with p vertices are also enumerated.

Isomorphism of circulant graphs and digraphs

Let S⊆ {1, …, n−1} satisfy − S = S mod n. The circulant graph G( n, S) with vertex set { v 0, v 1,…, v n−1 } and edge set E satisfies v i v j ϵ E if and only if j − i ∈ S, where all arithmetic is done mod n. The circulant digraph G( n, S) is defined similarly without the restriction S = − S. Ádám conjectured that G(n, S) ≊ G(n, S′) if and only if S = uS′ for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p 2, and determine exact conditions on S that it be true in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p 2 where p is an odd prime, or n is divisible by 2 4.