Regular Digraphs Research Articles

Network routing on regular digraphs and their line graphs

Network routing on regular digraphs and their line graphs

Hamilton cycles in dense regular digraphs and oriented graphs

We prove that for every ε>0 there exists n0=n0(ε) such that every regular oriented graph on n>n0 vertices and degree at least (1/4+ε)n has a Hamilton cycle. This establishes an approximate version of a conjecture of Jackson from 1981. We also establish a result related to a conjecture of Kühn and Osthus about the Hamiltonicity of regular directed graphs with suitable degree and connectivity conditions.

The Characteristic Polynomial of Sums of Random Permutations and Regular Digraphs

Abstract Let $A_{n}$ be the sum of $d$ permutation matrices of size $n\times n$, each drawn uniformly at random and independently. We prove that the normalized characteristic polynomial $\frac {1}{\sqrt {d}}\det (I_{n} - z A_{n}/\sqrt {d})$ converges when $n\to \infty $ towards a random analytic function on the unit disk. As an application, we obtain an elementary proof of the spectral gap of random regular digraphs. Our results are valid both in the regime where $d$ is fixed and for $d$ slowly growing with $n$.

Design of Coded Caching Schemes With Linear Subpacketizations Based on Injective Arc Coloring of Regular Digraphs

Coded caching is an effective technique to decongest the amount of traffic in the backhaul link. In such a scheme, each file hosted in the server is divided into a number of packets to pursue a low broadcasting rate based on the designed placements at each user’s cache. However, the implementation complexity of this scheme increases with the number of packets. It is important to design a scheme with a small subpacketization level and a relatively low transmission rate. Recently, placement delivery array (PDA) was proposed to address the subpacketization bottleneck of coded caching. This paper investigates the design of PDA from a new perspective, i.e., the injective arc coloring of regular digraphs. It is shown that the injective arc coloring of a regular digraph can yield a PDA with the same number of rows and columns. Based on this, a new class of regular digraphs are defined and the upper bounds on the injective chromatic index of such digraphs are derived. Consequently, four new coded caching schemes with a linear subpacketization level and a relatively small transmission rate are proposed, one of which generalizes the existing scheme for the scenario with a more flexible number of users.

Nonexistence of Almost Moore Digraphs of Degrees 4 and 5 with Self-Repeats

An almost Moore $(d,k)$-digraph is a regular digraph of degree $d>1$, diameter $k>1$ and order $N(d,k)=d+d^2+\cdots +d^k$. So far, their existence has only been shown for $k=2$, whilst it is known that there are no such digraphs for $k=3$, $4$ and for $d=2$, $3$ when $k\geq 3$. Furthermore, under certain assumptions, the nonexistence for the remaining cases has also been shown. In this paper, we prove that $(4,k)$ and $(5,k)$-almost Moore digraphs with self-repeats do not exist for $k\geq 5$.

Minimum arc-cuts of normally regular digraphs and Deza digraphs

A normally regular digraph with parameters (n,k,λ,μ) is a k-regular digraph on n vertices in which any two non-adjacent vertices have μ common out-neighbours, any two vertices connected by an arc in one direction have λ common out-neighbours and any two vertices connected by arcs in both directions have 2λ−μ common out-neighbours. A Deza digraph with parameters (n,k,b,c) is a k-regular digraph on n vertices such that any two distinct vertices have b or c common out-neighbours. In this paper, we determine the arc-connectivities and minimum arc-cuts of normally regular digraphs and Deza digraphs.

Some Constructions of Quasi-strongly Regular Digraphs

Some Constructions of Quasi-strongly Regular Digraphs

Majority Colorings of Sparse Digraphs

A majority coloring of a directed graph is a vertex-coloring in which every vertex has the same color as at most half of its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that every digraph has a majority $4$-coloring and conjectured that every digraph admits a majority $3$-coloring. They observed that the Local Lemma implies the conjecture for digraphs of large enough minimum out-degree if, crucially, the maximum in-degree is bounded by a(n exponential) function of the minimum out-degree.
 Our goal in this paper is to develop alternative methods that allow the verification of the conjecture for natural, broad digraph classes, without any restriction on the in-degrees. Among others, we prove the conjecture 1) for digraphs with chromatic number at most $6$ or dichromatic number at most $3$, and thus for all planar digraphs; and 2) for digraphs with maximum out-degree at most $4$. The benchmark case of $r$-regular digraphs remains open for $r \in [5,143]$.
 Our inductive proofs depend on loaded inductive statements about precoloring extensions of list-colorings. This approach also gives rise to stronger conclusions, involving the choosability version of majority coloring.
 We also give further evidence towards the existence of majority-$3$-colorings by showing that every digraph has a fractional majority 3.9602-coloring. Moreover we show that every digraph with large enough minimum out-degree has a fractional majority $(2+\varepsilon)$-coloring.

Vk- Super Vertex In-Magic Labelings of Directed Graphs

Let D be a digraph of order p and size q. For an integer k ⩾ 1 and ν ∈ V(D), let where Ek (v) is the set of all in-arcs which are at distance at most k from ν. A V k -super vertex in-magic labeling (Vk -SVIML) is an one-to-one onto function f : V(D) ∪ A(D) → {1,2…, P + q} such that f(V(D)) = {1,2…, p} and for every ν ∈ V(D), f(ν) + ω k(ν) = M for some positive integer M. A digraph that admits a V k -SVIML is called V k -super vertex in-magic (V k -SVIM). In this paper, we study some properties of V k -SVIML in digraphs. We characterized the digraphs which are V k -SVIM. Also, we find the magic constant for Ek -regular digraphs. Farther, we characterized the unidirectional cycles and union of unidirectional cycles which are V 2-SMM.

On the Aα-spectrum of joined union of digraphs

Let [Formula: see text] be a digraph of order [Formula: see text] and let [Formula: see text] be the adjacency matrix of [Formula: see text] Let Deg[Formula: see text] be the diagonal matrix of vertex out-degrees of [Formula: see text] For any real [Formula: see text] the generalized adjacency matrix [Formula: see text] of the digraph [Formula: see text] is defined as [Formula: see text] This matrix generalizes the spectral theories of the adjacency matrix and the signless Laplacian matrix of [Formula: see text]. In this paper, we find [Formula: see text]-spectrum of the joined union of digraphs in terms of spectrum of adjacency matrices of its components and the eigenvalues of an auxiliary matrix determined by the joined union. We determine the [Formula: see text]-spectrum of join of two regular digraphs and the join of a regular digraph with the union of two regular digraphs of distinct degrees. As applications, we obtain the [Formula: see text]-spectrum of various families of unsymmetric digraphs.

Lower bounds for the spectral norm of digraphs

Let D be a digraph with n vertices and σ1(D)≥σ2(D)≥⋯≥σn(D) the singular values of the adjacency matrix A of D. The spectral norm of D is σ1(D) and the trace norm of D is ‖D‖⁎=∑i=1nσi(D). In this paper we find lower bounds for the spectral norm of a digraph in terms of the structure of the digraph. Moreover, we introduce the concept of almost regular digraphs (extension of the well known almost regular graphs), and show that the lower bounds are attained precisely in almost regular digraphs. When we apply this theory to graphs, we recover well known lower bounds for the spectral radius of graphs. Also, we give a new upper bound for the trace norm of a digraph. Moreover, we determine the digraphs for which this bound is sharp: sink-source complete bipartite digraphs or symmetric balanced incomplete block designs (BIBD).

On the Dα spectral radius of strongly connected digraphs

Let G be a strongly connected digraph with distance matrix D(G) and let Tr(G) be the diagonal matrix with vertex transmissions of G. For any real ? ? [0, 1], define the matrix D?(G) as D?(G) = ?Tr(G) + (1-?)D(G). The D? spectral radius of G is the spectral radius of D?(G). In this paper, we first give some upper and lower bounds for the D? spectral radius of G and characterize the extremal digraphs. Moreover, for digraphs that are not transmission regular, we give a lower bound on the difference between the maximum vertex transmission and the D? spectral radius. Finally, we obtain the D? eigenvalues of the join of certain regular digraphs.

Spectra of products of digraphs

A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the new distance matrix, with natural extensions to the distance Laplacian and distance signless Laplacian, in addition to the new adjacency matrix, with natural extensions to the Laplacian and signless Laplacian. Various sums of Kronecker products of nonnegative matrices are introduced to model the Cartesian and lexicographic products of digraphs. The Jordan canonical form is applied extensively to the analysis of spectra and eigenvectors. The analysis shows that Cartesian products provide a method for building infinite families of transmission regular digraphs with few distinct distance eigenvalues.

On the counting problem in inverse Littlewood–Offord theory

Let ε 1 , … , ε n be independent and identically distributed Rademacher random variables taking values ± 1 with probability 1 / 2 each. Given an integer vector a = ( a 1 , … , a n ) , its concentration probability is the quantity ρ ( a ) : = sup x ∈ Z Pr ( ε 1 a 1 + ⋯ + ε n a n = x ) . The Littlewood–Offord problem asks for bounds on ρ ( a ) under various hypotheses on a, whereas the inverse Littlewood–Offord problem, posed by Tao and Vu, asks for a characterization of all vectors a for which ρ ( a ) is large. In this paper, we study the associated counting problem: How many integer vectors a belonging to a specified set have large ρ ( a ) ? The motivation for our study is that in typical applications, the inverse Littlewood–Offord theorems are only used to obtain such counting estimates. Using a more direct approach, we obtain significantly better bounds for this problem than those obtained using the inverse Littlewood–Offord theorems of Tao and Vu and of Nguyen and Vu. Moreover, we develop a framework for deriving upper bounds on the probability of singularity of random discrete matrices that utilizes our counting result. To illustrate the methods, we present the first ‘exponential-type’ (that is, exp ( − c n c ) for some positive constant c) upper bounds on the singularity probability for the following two models: (i) adjacency matrices of dense signed random regular digraphs, for which the previous best-known bound is O ( n − 1 / 4 ) , due to Cook; and (ii) dense row-regular { 0 , 1 } -matrices, for which the previous best-known bound is O C ( n − C ) for any constant C > 0 , due to Nguyen.

The Circular Law for random regular digraphs

Soit $\log^{C}n\le d\le n/2$ pour une constante suffisamment grande $C>0$. Notons $A_{n}$ la matrice d’adjacence d’un graphe dirigé aléatoire $d$-régulier sur $n$ sommets. Nous montrons que lorsque $n$ tend vers l’infini, la distribution empirique des valeurs propres de $A_{n}$, convenablement normalisée, suit la loi du cercle. Une étape cruciale consiste à obtenir une borne inférieure quantitative asymptotique pour la plus petite valeur singulière de perturbations additives de $A_{n}$.

Structure of eigenvectors of random regular digraphs

Let $n$ be a large integer, let $d$ satisfy $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in {\mathcal C}$. Further, denote by $M$ the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. In this paper, we study structure of the kernel of submatrices of $M-z\,{\rm Id}$, formed by removing a subset of rows. We show that with large probability the kernel consists of two non-intersecting types of vectors, which we call very steep and sloping with many levels. As a corollary of this theorem, we show that every eigenvector of $M$, except for constant multiples of $(1,1,\dots,1)$, possesses a weak delocalization property: all subsets of coordinates with the same value have cardinality less than $Cn\ln^2 d/\ln n$. For a large constant $d$ this provides a principally new structural information on eigenvectors, implying that the number of their level sets grows to infinity with $n$, and -- combined with additional arguments -- that the rank of the adjacency matrix $M$ is at least $n-1$, with probability going to 1 as $n\to\infty$. Moreover, our main result for $d$ slowly growing to infinity with $n$ contributes a crucial step towards establishing the circular law for the empirical spectral distribution of sparse random $d$-regular directed graphs. As a key technical device we introduce a decomposition of ${\mathcal C}^n$ into vectors of different degree of structuredness, which is an alternative to the decomposition based on the least common denominator in the regime when the underlying random matrix is very sparse.

Arc fault tolerance of Cartesian product of regular digraphs on super-restricted arc-connectivity

Arc fault tolerance of Cartesian product of regular digraphs on super-restricted arc-connectivity

A spectral excess theorem for digraphs with normal Laplacian matrices

The spectral excess theorem‎, ‎due to Fiol and Garriga in 1997‎, ‎is an important result‎, ‎because it gives a good characterization‎ ‎of distance-regularity in graphs‎. ‎Up to now‎, ‎some authors have given some variations of this theorem‎. ‎Motivated by this‎, ‎we give the corresponding result by using the Laplacian spectrum for digraphs‎. ‎We also illustrate this Laplacian spectral excess theorem for digraphs with few Laplacian eigenvalues and we show that any strongly connected and regular digraph that has normal Laplacian matrix with three distinct eigenvalues‎, ‎is distance-regular‎. ‎Hence such a digraph is strongly regular with girth $g=2$ or $g=3$‎.

The rank of random regular digraphs of constant degree

Let d be a (large) integer. Given n≥2d, let An be the adjacency matrix of a random directed d-regular graph on n vertices, with the uniform distribution. We show that the rank of An is at least n−1 with probability going to one as n grows to infinity. The proof combines the well known method of simple switchings and a recent result of the authors on delocalization of eigenvectors of An.

Upper bound for the trace norm of the Laplacian matrix of a digraph and normally regular digraphs

The trace norm of M∈Mn(C) is defined as ‖M‖⁎=∑k=1nσk, where σ1≥σ2≥⋯≥σn≥0 are the singular values of M (i.e. the square roots of the eigenvalues of MM⁎). We are particularly interested in the trace norm ‖L(D)−anIn‖⁎, where L(D) is the Laplacian matrix of a digraph D with n vertices and a arcs, and In is the n×n identity matrix. When D=G is a graph with n vertices and m edges, then ‖L(D)−anIn‖⁎=‖L(G)−2mnIn‖⁎=LE(G), the Laplacian energy of G introduced by Gutman and Zhou in 2006. We show that for a digraph D with n vertices and a arcs,‖L(D)−anIn‖⁎≤n(a−a2n+∑i=1n(di+)2), where d1+,…,dn+ are the outer degrees of the vertices of D. Moreover, the digraphs where this bound is attained are special classes of normally regular digraphs studied by Jørgensen in 2015 [6]. Finally, we construct normally regular digraphs where the equality is attained.