Family Of Digraphs Research Articles

The birth of the strong components

AbstractIt is known that random directed graphs undergo a phase transition around the point . Earlier, Łuczak and Seierstad have established that as when , the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as goes from to . By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of and provide more statistical insights into the structure of a random digraph around, below and above its transition point. We obtain the limiting probability that a random digraph is acyclic and the probability that it has one strongly connected complex component with a given difference between the number of edges and vertices (called excess). Our result can be extended to the case of several complex components with given excesses as well in the whole range of sparse digraphs. Our study is based on a general symbolic method which can deal with a great variety of possible digraph families, and a version of the saddle point method which can be systematically applied to the complex contour integrals appearing from the symbolic method. While the technically easiest model is the model of random multidigraphs, in which multiple edges are allowed, and where edge multiplicities are sampled independently according to a Poisson distribution with a fixed parameter , we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2‐cycles are either allowed or not. Our theoretical predictions are supported by numerical simulations when the number of vertices is finite, and we provide tables of numerical values for the integrals of Airy functions that appear in this study.

Semicomplete compositions of digraphs

Let T be a digraph with vertices u1,…,ut (t≥2), and let H1,…,Ht be digraphs such that Hi has vertices ui,ji, where 1≤ji≤ni. The composition Q=T[H1,…,Ht] is a digraph with vertex setV(Q)={ui,ji:1≤i≤t,1≤ji≤ni} and arc set(⋃i=1tA(Hi))⋃(⋃uiup∈A(T){ui,jiup,qp:1≤ji≤ni,1≤qp≤np}). The composition Q=T[H1,…,Ht] is a semicomplete composition if T is semicomplete, i.e., there is at least one arc between every pair of vertices. Digraph compositions generalize some families of digraphs, including quasi-transitive digraphs, extended semicomplete digraphs and lexicographic product digraphs. In particular, strong semicomplete compositions form a significant generalization of strong quasi-transitive digraphs.In this paper, we study the structural properties of semicomplete compositions and obtain results on connectivity, paths, cycles and strong spanning subdigraphs. Our results show that this class of digraphs shares some nice properties of quasi-transitive digraphs.

On the Laplacian spread of digraphs

In this article, we extend the notion of the Laplacian spread to simple directed graphs (digraphs) using the restricted numerical range. First, we provide Laplacian spread values for several families of digraphs. Then, we prove sharp upper bounds on the Laplacian spread for all polygonal and balanced digraphs. In particular, we show that the validity of the Laplacian spread bound for balanced digraphs is equivalent to the Laplacian spread conjecture for simple undirected graphs, which was conjectured in 2011 and proven in 2021. Moreover, we prove an equivalent statement for weighted balanced digraphs with weights between 0 and 1. Finally, we state several open conjectures that are motivated by empirical data.

Small Mahler Measures From Digraphs

We attach Mahler measures to digraphs and find combinatorial realizations of nearly all of the known low-degree ( ) small (< 1.3) one-variable Mahler measures. We find one new such measure not on either of the lists maintained by Mossinghoff and Sac-Épée. Considering limits of sequences of measures attached to families of digraphs, we get combinatorial explanations for 57 of the 61 known irreducible two-variable measures below 1.37.

On the chromatic number of the preferential attachment graph

We prove that for every m∈N and every δ∈(−m,0), the chromatic number of the preferential attachment graph PAt(m,δ) is asymptotically almost surely equal to m+1. The proof relies on a combinatorial construction of a family of digraphs of chromatic number m+1 followed by a proof that asymptotically almost surely there is a digraph in this family, which is realised as a subgraph of the preferential attachment graph.

Associative spectra of graph algebras I

Associative spectra of graph algebras are examined with the help of homomorphisms of DFS trees. Undirected graphs are classified according to the associative spectra of their graph algebras; there are only three distinct possibilities: constant 1, powers of 2, and Catalan numbers. Associative and antiassociative digraphs are described, and associative spectra are determined for certain families of digraphs, such as paths, cycles, and graphs on two vertices.

Seymour’s Second Neighborhood Conjecture for 6-antitransitive digraphs

Seymour’s Second Neighborhood Conjecture states that every simple oriented graph has a vertex such that the cardinality of its second neighborhood is greater than or equal to the cardinality of its first neighborhood. The conjecture has been shown to hold for various families of digraphs but remains unsettled. A digraph is said to be k-antitransitive if any u to v path of length k implies there is no u to v edge. If the conjecture is shown to hold for k-antitransitive digraphs for an arbitrary k, this would settle it for finite digraphs, as every finite digraph is k-antitransitive for k greater than the length of its longest path. So far, the conjecture has been shown to hold for k-antitransitive simple oriented graphs for k≤5. In this paper we prove the conjecture for 6-antitransitive simple oriented graphs.

Antimagic Naming of Specific Groups of Digraphs Utilizing Snared Skolem Arrangement

The vertex sum at each given vertex u is equal to the total of all the edges directed into it minus the sum of all the edges directed out of it. This is known as an anti magicl abeling of a directed graph G with v vertices and q edges. An antimagic digraph is a digraph that permits an antimagiclabeling. In this paper, we analyze a few antimagic digraph features and use hooked Skolem sequences to demonstrate the anti magicness of specific digraph families.

On skew Laplacian spectrum and energy of digraphs

We consider the skew Laplacian matrix of a digraph [Formula: see text] obtained by giving an arbitrary direction to the edges of a graph [Formula: see text] having [Formula: see text] vertices and [Formula: see text] edges. With [Formula: see text] to be the skew Laplacian eigenvalues of [Formula: see text], the skew Laplacian energy [Formula: see text] of [Formula: see text] is defined as [Formula: see text]. In this paper, we analyze the effect of changing the orientation of an induced subdigraph on the skew Laplacian spectrum. We obtain bounds for the skew Laplacian energy [Formula: see text] in terms of various parameters associated with the digraph [Formula: see text] and the underlying graph [Formula: see text] and we characterize the extremal digraphs attaining these bounds. We also show these bounds improve some known bounds for some families of digraphs. Further, we show the existence of some families of skew Laplacian equienergetic digraphs.

Classification using proximity catch digraphs

We employ random geometric digraphs to construct semi-parametric classifiers. These data-random digraphs belong to parameterized random digraph families called proximity catch digraphs (PCDs). A related geometric digraph family, class cover catch digraph (CCCD), has been used to solve the class cover problem by using its approximate minimum dominating set and showed relatively good performance in the classification of imbalanced data sets. Although CCCDs have a convenient construction in $${\mathbb {R}}^d$$, finding their minimum dominating sets is NP-hard and their probabilistic behaviour is not mathematically tractable except for $$d=1$$. On the other hand, a particular family of PCDs, called proportional-edge PCDs (PE-PCDs), has mathematically tractable minimum dominating sets in $${\mathbb {R}}^d$$; however their construction in higher dimensions may be computationally demanding. More specifically, we show that the classifiers based on PE-PCDs are prototype-based classifiers such that the exact minimum number of prototypes (equivalent to minimum dominating sets) is found in polynomial time on the number of observations. We construct two types of classifiers based on PE-PCDs. One is a family of hybrid classifiers that depends on the location of the points of the training data set, and another type is a family of classifiers solely based on class covers. We assess the classification performance of our PE-PCD based classifiers by extensive Monte Carlo simulations, and compare them with that of other commonly used classifiers. We also show that, similar to CCCD classifiers, our classifiers tend to be robust to the class imbalance in classification as well.

Isometric copies of directed trees in orientations of graphs

AbstractThe isometric Ramsey number of a family of digraphs is the smallest number of vertices in a graph such that any orientation of the edges of contains every member of in the distance‐preserving way. We observe that the isometric Ramsey number of a finite family of finite acyclic digraphs is always finite, and present some bounds in special cases. For example, we show that the isometric Ramsey number of the family of all oriented trees with vertices is at most .

Characterization of color patterns by dynamic [formula omitted]-paths

Let H be a digraph possibly with loops, let D be a multidigraph, and let c:A(D)→V(H) be a coloring of the arcs of D with the vertices of H. A sequence W of vertices in D, W=(x0,x1,…,xn), is a dynamic H-walk if for each i∈{0,…,n−2} there exist fi and fi+1 such that fi is an arc from xi to xi+1, fi+1 is an arc from xi+1 to xi+2, and (c(fi),c(fi+1)) is an arc of H. If W does not repeat vertices, W is a dynamic H-path. For u,v∈V(D), we say that u reaches v by dynamic H-paths if there exists a dynamic H-path from u to v in D. A subset K⊆V(D) is a kernel by dynamic H-paths of D if every vertex in V(D)−K reaches some vertex in K by dynamic H-paths (absorbent by dynamic H-paths), and no vertex in K can be reached by another vertex in K by dynamic H-paths in D (independent by dynamic H-paths).Let D̃3 be the family of digraphs H such that for every H-arc-colored multidigraph D, D has a kernel by dynamic H-paths. Let D̃2 (resp. D̃1) be the class of digraphs H such that for every H-arc-colored multidigraph D (resp. multitournament), D has an independent absorbent by dynamic H-paths set. In this work, we provide a characterization of D̃2 and show that D̃3 is one of two families.

Critical Kernel Imperfect Problem in Generalizations of Bipartite Tournaments

Kernel is an important topic in digraphs. A digraph such that every proper induced subdigraph has a kernel is said to be critical kernel imperfect (CKI, for short) if the digraph does not have a kernel. Galeana-Sanchez and Olsen characterized the CKI-digraphs for the following families of digraphs: asymmetric arc-locally in-/out-semicomplete digraphs, asymmetric 3-quasi-transitive digraphs and asymmetric 3-anti-quasi-transitive \(TT_3\)-free digraphs. In this paper, we shall completely characterize the above four classes CKI-digraphs without any restriction on arcs and \(TT_3\)-free subdigraphs.

Quasi-thin weakly distance-regular digraphs

A weakly distance-regular digraph is quasi-thin if the maximum value of its intersection numbers is 2. In this paper, we focus on commutative quasi-thin weakly distance-regular digraphs, and classify such digraphs with valency more than 3. As a result, this family of digraphs is completely determined.

An algebraic approach to lifts of digraphs

We present some applications of a new matrix approach for studying the properties of the lift Γα of a voltage digraph, which has arcs weighted by the elements of a group. As a main result, when the involved group is Abelian, we completely determine the spectrum of Γα. As some examples of our technique, we study some basic properties of the Alegre digraph, and completely characterize the spectrum of a new family of digraphs, which contains the generalized Petersen graphs, and the Hoffman–Singleton graph.

Zero forcing in iterated line digraphs

Zero forcing is a propagation process on a graph, or digraph, defined in linear algebra to provide a bound for the minimum rank problem. Independently, zero forcing was introduced in physics, computer science and network science, areas where line digraphs are frequently used as models. Zero forcing is also related to power domination, a propagation process that models the monitoring of electrical power networks.In this paper we study zero forcing in iterated line digraphs and provide a relationship between zero forcing and power domination in line digraphs. In particular, for regular iterated line digraphs we determine the minimum rank/maximum nullity, zero forcing number and power domination number, and provide constructions to attain them. We conclude that regular iterated line digraphs present optimal minimum rank/maximum nullity, zero forcing number and power domination number, and apply our results to determine those parameters on some families of digraphs often used in applications.

Asymptotically optimal amplifiers for the Moran process

We study the Moran process as adapted by Lieberman, Hauert and Nowak. This is a model of an evolving population on a graph or digraph where certain individuals, called “mutants” have fitness r and other individuals, called “non-mutants” have fitness 1. We focus on the situation where the mutation is advantageous, in the sense that r>1. A family of digraphs is said to be strongly amplifying if the extinction probability tends to 0 when the Moran process is run on digraphs in this family. The most-amplifying known family of digraphs is the family of megastars of Galanis et al. We show that this family is optimal, up to logarithmic factors, since every strongly-connected n-vertex digraph has extinction probability Ω(n−1/2). Next, we show that there is an infinite family of undirected graphs, called dense incubators, whose extinction probability is O(n−1/3). We show that this is optimal, up to constant factors. Finally, we introduce sparse incubators, for varying edge density, and show that the extinction probability of these graphs is O(n/m), where m is the number of edges. Again, we show that this is optimal, up to constant factors.

Pairings and related symmetry notions

In this paper, we give a purely mathematical generalization of an information table. We call pairing on a given set $$\Omega $$ a triple $$\mathfrak {P}=(U, F, \Lambda )$$ , where U and $$\Lambda $$ are non-empty sets and $$F:U\times \Omega \rightarrow \Lambda $$ is a map. We provide several examples of pairings: graphs, digraphs, metric spaces, group actions and vector spaces endowed with a bilinear form. Moreover, we reinterpret the usual notion of indiscernibility (with respect to a fixed attribute subset of an information table) in terms of local symmetry on U and, then, we study a global version of symmetry, that we called indistinguishability. In particular, we interpret the latter relation as the symmetrization of a pre-order $$\le _{\mathfrak {P}}$$ , that describes the symmetry transmission between subsets of $$\Omega $$ . Hence, we introduce a global average of symmetry transmission and studied it for some basic digraph families. Finally, we prove that the partial order of any finite lattice can be described in terms of the above pre-order.

A note on a directed version of the 1-2-3 Conjecture

The least k such that a given digraph D=(V,A) can be arc-labeled with integers in the interval [1,k] so that the sum of values in-coming to x is distinct from the sum of values out-going from y for every arc (x,y)∈A, is denoted by χ̄Łe(D). This corresponds to one of possible directed versions of the well-known 1-2-3 Conjecture. Unlike in the case of other possibilities, we show that χ̄Łe(D) is unbounded in the family of digraphs for which this parameter is well defined. However, if the family is restricted by excluding the digraphs with so-called lonely arcs, we prove that χ̄Łe(D)≤4, and we conjecture that χ̄Łe(D)≤3 should hold.

Dependency and accuracy measures for directed graphs

In this paper we use finite directed graphs (digraphs) as mathematical models to study two basic notions widely analyzed in granular computing: the attribute dependency and the approximation accuracy. To be more specific, at first we interpret any digraph as a Boolean information table, next we study the approximation accuracy for three fundamentals digraph families: the directed path, the directed cycle and the transitive tournament. We also introduce a new global average for the attribute dependency in any information table and we determine such number for any directed path. For the transitive tournament we provide a lower bound.