Circulant Digraphs Research Articles

Bounds on the acyclic disconnection of a digraph

The acyclic disconnectionω→(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overrightarrow{\\omega }(D)$$\\end{document} of a digraph D is the maximum possible number of (weakly) connected components of a digraph obtained from D by deleting an acyclic set of arcs. In this paper, we provide new lower and upper bounds in terms of properties such as the degree, the directed girth, and the existence of certain subdigraphs and bounds for bipartite digraphs, p-cycles, and some circulant digraphs. Finally, as a consequence of our bounds, we prove the Conjecture of Caccetta and Häggkvist for a particular class of digraphs.

Necessary and Sufficient Conditions for Circulant Digraphs to be Antistrong, Weakly-Antistrong and Anti-Eulerian

Necessary and Sufficient Conditions for Circulant Digraphs to be Antistrong, Weakly-Antistrong and Anti-Eulerian

Correction to “A Classification of Spectrum-determined Circulant Digraphs”

Correction to “A Classification of Spectrum-determined Circulant Digraphs”

Proper connection and proper-walk connection of digraphs

An arc-colored digraph D is properly (properly-walk) connected if, for any ordered pair of vertices (u,v), the digraph D contains a directed path (a directed walk) from u to v such that arcs adjacent on that path (on that walk) have distinct colors. The proper connection number pc→(D) (the proper-walk connection number wc→(D)) of a digraph D is the minimum number of colours to make D properly connected (properly-walk connected). We prove that pc→(Cn(S))≤2 for every circulant digraph Cn(S) with S⊆{1,…,n−1},|S|≥2 and 1∈S. Furthermore, we give some sufficient conditions for a Hamiltonian digraph D to satisfy pc→(D)=wc→(D)=2.

(Strong) Total proper connection of some digraphs

The total proper connection number of a given digraph D, represented by $$\overrightarrow{tpc}(D)$$ , denotes the smallest number of colors needed for making D total proper connected. The strong total proper connection number of D, represented by $$\overrightarrow{stpc}(D)$$ , shows the smallest number of colors required for making D strong total proper connected. In the present work, we represent some preliminary findings on $$\overrightarrow{tpc}(D)$$ and $$\overrightarrow{stpc}(D)$$ . Moreover, findings on the (strong) total proper connection numbers of biorientations of graphs, circle digraphs, circulant digraphs and cacti digraphs are provided.

Hyper-Hamiltonian circulants

A Hamiltonian graph G = ( V,E ) is called hyper-Hamiltonian if G - v is Hamiltonian for any v ∈ V ( G ). G is called a circulant if its automorphism group contains a | V ( G )|-cycle. First, we give the necessary and sufficient conditions for any undirected connected circulant to be hyper-Hamiltonian. Second, we give necessary and sufficient conditions for a connected circulant digraph with two jumps to be hyper-Hamiltonian. In addition, we specify some sufficient conditions for a circulant digraph with arbitrary number of jumps to be hyper-Hamiltonian.

(Strong) Proper Connection in Some Digraphs

An arc-colored digraph D is proper connected if any pair of vertices vi, vj e V(D) there is a proper vi - vj path whose adjacent arcs have different colors and a proper vj - vi path whose adjacent arcs have different colors. The proper connection number of a digraph D is the minimum number of colors needed to make D proper connected, denoted by spc(D). An arc-colored digraph D is strong proper connected if any pair of vertices vi, vj e V(D) there is a proper vi - vj geodesic and a proper vj - vi geodesic. The strong proper connection number of D is the minimum number of colors required to color the arcs of D in order to make D strong proper connected, denoted by spc(D). In this paper, we will show some results on spc(D) and spc(D), mostly for the case of the (strong) proper connection numbers of cacti and circulant digraphs.

A Classification of Spectrum-determined Circulant Digraphs

Some researchers have proved that adam’s conjecture is wrong. However, under special conditions, it is right. Let Zn be a cyclic group of order n and Cn(S) be the circulant digraph of Zn with respect to S ⊆ Zn\{0}. In the literature, some people have used a spectral method to solve the isomorphism for the circulants of prime-power order. In this paper, we also use the spectral method to characterize the circulants of order paqbwc(where p, q and w are all distinct primes), and to make adam’s conjecture right.

Rainbow vertex connection of digraphs

An edge-coloured path is rainbow if its edges have distinct colours. An edge-coloured connected graph is said to be rainbow connected if any two vertices are connected by a rainbow path, and strongly rainbow connected if any two vertices are connected by a rainbow geodesic. The (strong) rainbow connection number of a connected graph is the minimum number of colours needed to make the graph (strongly) rainbow connected. These two graph parameters were introduced by Chartrand et al. (Math Bohem 133:85–98, 2008). As an extension, Krivelevich and Yuster proposed the concept of rainbow vertex-connection. The topic of rainbow connection in graphs drew much attention and various similar parameters were introduced, mostly dealing with undirected graphs. Dorbec, Schiermeyer, Sidorowicz and Sopena extended the concept of the rainbow connection to digraphs. In this paper, we consider the (strong) rainbow vertex-connection number of digraphs. Results on the (strong) rainbow vertex-connection number of biorientations of graphs, cycle digraphs, circulant digraphs and tournaments are presented.

Circulant Homogeneous Factorisations of Complete Digraphs $\rm\bf K_{p^d}$ with $p$ an Odd Prime

Let $\mathcal F=(\rm\bf K_{n},\mathcal P)$ be a circulant homogeneous factorisation of index $k$, that means $\mathcal P$ is a partition of the arc set of the complete digraph $\rm\bf K_n$ into $k$ circulant factor digraphs such that there exists $\sigma\in S_n$ permuting the factor circulants transitively amongst themselves. Suppose further such an element $\sigma$ normalises the cyclic regular automorphism group of these circulant factor digraphs, we say $\mathcal F$ is normal. Let $\mathcal F=(\rm\bf K_{p^d},\mathcal P)$ be a circulant homogeneous factorisation of index $k$ where $p^d$, ($d\ge 1$) is an odd prime power. It is shown in this paper that either $\mathcal F$ is normal or $\mathcal F$ is a lexicographic product of two smaller circulant homogeneous factorisations.

Automorphism Groups of Circulant Digraphs With Applications to Semigroup Theory

We characterize the automorphism groups of circulant digraphs whose connection sets are relatively small, and of unit circulant digraphs. For each class, we either explicitly determine the automorphism group or we show that the graph is a “normal” circulant, so the automorphism group is contained in the normalizer of a cycle. Then we use these characterizations to prove results on the automorphisms of the endomorphism monoids of those digraphs. The paper ends with a list of open problems on graphs, number theory, groups and semigroups.

Rainbow Connection in Some Digraphs

An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. This concept was introduced by Chartrand et al. (Math Bohemica 133(1):85---98, 2008), and it was extended to oriented graphs by Dorbec et al. (Discrete Appl Math 179(31):69---78, 2014). In this paper we present some results regarding this extension, mostly for the case of circulant digraphs.

Self-diclique circulant digraphs

We study a particular digraph dynamical system, the so called digraph diclique operator. Dicliques have frequently appeared in the literature the last years in connection with the construction and analysis of different types of networks, for instance biochemical, neural, ecological, sociological and computer networks among others. Let $D=(V,A)$ be a reflexive digraph (or network). Consider $X$ and $Y$ (not necessarily disjoint) nonempty subsets of vertices (or nodes) of $D$. A disimplex $K(X,Y)$ of $D$ is the subdigraph of $D$ with vertex set $X\cup Y$ and arc set $\{(x,y)\colon x\in X, y\in Y\}$ (when $X\cap Y\neq \varnothing $, loops are not considered). A disimplex $K(X,Y)$ of $D$ is called a diclique of $D$ if $K(X,Y)$ is not a proper subdigraph of any other disimplex of $D$. The diclique digraph $\overrightarrow {k}(D)$ of a digraph $D$ is the digraph whose vertex set is the set of all dicliques of $D$ and $( K(X,Y),K(X',Y'))$ is an arc of $\overrightarrow {k}(D)$ if and only if $Y\cap X'\neq \varnothing $. We say that a digraph $D$ is self-diclique if $\overrightarrow {k}(D)$ is isomorphic to $D$. In this paper, we provide a characterization of the self-diclique circulant digraphs and an infinite family of non-circulant self-diclique digraphs.

Strongly regular m-Cayley circulant graphs and digraphs

The first part of this paper is a survey about strongly regular graphs and digraphs admitting a semiregular cyclic group of automorphisms. In the second part, some new types of such digraphs, called uniform and almost uniform, are studied. By using partial sum families, the form of the parameters is determined and some directed strongly regular graphs derived from these partial sum families with previously unknown parameters are obtained.

On the automorphism groups of almost all circulant graphs and digraphs

We attempt to determine the structure of the automorphism group of a generic circulant graph. We first show that almost all circulant graphs have automorphism groups as small as possible. The second author has conjectured that almost all of the remaining circulant (di)graphs (those whose automorphism group is not as small as possible) are normal circulant (di)graphs. We show this conjecture is not true in general, but is true if we consider only those circulant (di)graphs whose order is in a "large" subset of integers. We note that all non-normal circulant (di)graphs can be classified into two natural classes (generalized wreath products, and deleted wreath type), and show that neither of these classes contains almost every non-normal circulant digraph.

Decomposition of circulant digraphs with two jumps into cycles of equal lengths

Let G=Gn(a1,a2) be a connected circulant digraph of order n with two distinct jumps a1,a2<n. We give several sufficient conditions for a decomposition of Gn(a1,a2) into directed cycles of equal lengths. We then prove that Gn(a1,a2) contains a 2-factor consisting of all cycles of equal lengths and comprised of both jumps if and only if gcd(n,s1a1+s2a2)=k(s1+s2) and a1≡a2(mods1+s2) for some positive integers k,s1,s2. Based on this last result we prove that Gn(a1,a2) can be decomposed into two 2-factors with all cycles comprising both jumps if and only if gcd(n,s1a1+s2a2)=gcd(n,s1a2+s2a1)=k(s1+s2) and a1≡a2(mods1+s2) for some positive integers k,s1,s2. Furthermore, we prove that if such a decomposition exists then all resulting cycles are of equal lengths.

Kernels in circulant digraphs

A kernel $J$ of a digraph $D$ is an independent set of vertices of $D$ such that for every vertex $w,in,V(D),setminus,J$ there exists an arc from $w$ to a vertex in $J.$‎ ‎In this paper‎, ‎among other results‎, ‎a characterization of $2$-regular circulant digraph having a kernel is obtained‎. ‎This characterization is a partial solution to the following problem‎: ‎Characterize circulant digraphs which have kernels; it appeared in the book Digraphs‎ - ‎theory‎, ‎algorithms and applications‎, ‎Second Edition‎, ‎Springer-Verlag‎, ‎2009‎, ‎by J‎. ‎Bang-Jensen and G‎. ‎Gutin‎.

Characteristic Polynomials and Spectra of Some Block Circulant Graphs

A block matrix A=(Aij ) n×n is called a block circulant matrix, denoted by C(A 11, A 12, …, A 1n ) if Aij =A 1,j−i+1 for all 1⩽i, j⩽n, where Aij is a matrix and j−i+1 is taken modulo n. Especially when all Aij are of order 1, A is called a circulant matrix. A digraph is called a (block) circulant digraph if its adjacency matrix is congruent to a (block) circulant matrix. In this article we will calculate the characteristic polynomials and spectra of several special block circulant graphs having some physical and chemical backgrounds.

Circulant digraphs integral over number fields

A number field K is a finite extension of rational number field Q. A circulant digraph integral over K means that all its eigenvalues are algebraic integers of K. In this paper we give the necessary and sufficient condition for circulant digraphs which are integral over a number field K. We also solve Conjecture 3.3 in Xu and Meng (2011) [11] and find that it is affirmative.

A Note on Hamiltonian Circulant Digraphs of Outdegree Three

We construct Hamilton cycles in connected loopless circulant digraphs of outdegree three with connection set of the form for an integer satisfying the condition for some integer such that , where . This extends work of Miklavi and ?parl, who previously deter-mined the Hamiltonicity of these digraphs in the case where and , to other values of which depend on the generators and .