The spectra of Manhattan street networks

The multidimensional Manhattan street networks constitute a family of digraphs with many interesting properties, such as vertex symmetry (in fact they are Cayley digraphs), easy routing, Hamiltonicity, and modular structure. From the known structural properties of these digraphs, we determine their spectra, which always contain the spectra of hypercubes. In particular, in the standard (two-dimensional) case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity.     

On the weak distance-regularity of Moore-type digraphs

We prove that Moore digraphs, and some other classes of extremal digraphs, are weakly distance-regular in the sense that there is an invariance of the number of walks between vertices at a given distance. As weakly distance-regular digraphs, we then compute their complete spectrum from a ‘small’ intersection matrix. This is a very useful tool for deriving some results about their existence and/or their structural properties. For instance, we present here an alternative and unified proof of the existence results on Moore digraphs, Moore bipartite digraphs and, more generally, Moore generalized p-cycles. In addition, we show that the line digraph structure appears as a characteristic property of any Moore generalized p-cycle of diameter D?≥?2p.     

On the structure of locally semicomplete digraphs

In this paper we continue the study, started by J. Bang-Jensen (1989), of locally semicomplete digraphs, a generalization of tournaments, to which many well-known tournament results extend. The underlying undirected graphs of the locally semicomplete digraphs are precisely the proper circular-arc graphs. We give new results on the structure of locally semicomplete digraphs, as well as several examples of properties of tournaments and semicomplete digraphs that do not extend to the class of locally semicomplete digraphs.     

Ranked Highly-Arc-Transitive Digraphs as Colimits

This paper builds on work characterising digraphs with rich transitivity properties. When such digraphs are equipped with a rank function onto a colexicographic power of ? they arise as direct limits of better understood structures. These structures, it will be seen, contain sufficient information to describe the cycles of the full structure.     

Digraphs with real and gaussian spectra

The conventional binary operations of cartesian product, conjunction, and composition of two digraphs D₁ and D₂ are observed to give the sum, the product, and a more complicated combination of the spectra of D₁ and D₂ as the resulting spectrum. These formulas for analyzing the spectrum of a digraph are utilized to construct for any positive integer n, a collection of n nonisomorphic strong regular nonsymmetric digraphs with real spectra. Further, an infinite collection of strong nonsymmetric digraphs with nonzero gaussian integer value is found. Finally, for any n, it is shown that there are n cospectral strong nonsymmetric digraphs with integral spectra.     

Weakly distance-regular digraphs

We introduce the concept of weakly distance-regular digraph and study some of its basic properties. In particular, the (standard) distance-regular digraphs, introduced by Damerell, turn out to be those weakly distance-regular digraphs which have a normal adjacency matrix. As happens in the case of distance-regular graphs, the study is greatly facilitated by a family of orthogonal polynomials called the distance polynomials. For instance, these polynomials are used to derive the spectrum of a weakly distance-regular digraph. Some examples of these digraphs, such as the butterfly and the cycle prefix digraph which are interesting for their applications, are analyzed in the light of the developed theory. Also, some new constructions involving the line digraph and other techniques are presented.     

Quasi-transitive digraphs

A digraph is quasi-transitive if there is a complete adjacency between the inset and the outset of each vertex. Quasi-transitive digraphs are interseting because of their relation to comparability graphs. Specifically, a graph can be oriented as a quasi-transitive digraph if and only if it is a comparability graph. Quasi-transitive digraphs are also of interest as they share many nice properties of tournaments. Indeed, we show that every strongly connected quasi-transitive digraphs D on at least four vertices has two vertices v₁ and v₂ such that D – v_i is strongly connected for i = 1, 2. A result of tournaments on the existence of a pair of arc-disjoint in- and out-branchings rooted at the same vertex can also be extended to quasi-transitive digraphs. However, some properties of tournaments, like hamiltonicity, cannot be extended directly to quasi-transitive digraphs. Therefore we characterize those quasi-transitive digraphs which have a hamiltonian cycle, respectively a hamiltonian path. We show the existence of highly connected quasi-transitive digraphs D with a factor (a collection of disjoint cycles covering the vertex set of D), which have a cycle of every length 3 ≦ k ≦ |V(D)| ? 1 through every vertex and yet they are not hamiltonian. Finally we characterize pancyclic and vertex pancyclic quasi-transitive digraphs. © 1995, John Wiley & Sons, Inc.     

Making choices with a binary relation: Relative choice axioms and transitive closures

This article presents an axiomatic analysis of the best choice decision problem from a reflexive crisp binary relation on a finite set (a digraph). With respect to a transitive digraph, optimality and maximality are usually accepted as the best fitted choice axioms to the intuitive notion of best choice. However, beyond transitivity (resp. acyclicity), optimality and maximality can characterise distinct choice sets (resp. empty sets). Accordingly, different and rather unsatisfying concepts have appeared, such as von Neumann–Morgenstern domination, weak transitive closure and kernels. Here, we investigate a new family of eight choice axioms for digraphs: relative choice axioms. Within choice theory, these axioms generalise top-cycle for tournaments, gocha, getcha and rational top-cycle for complete digraphs. We present their main properties such as existence, uniqueness, idempotence, internal structure, and cross comparison. We then show their strong relationship with optimality and maximality when the latter are not empty. Otherwise, these axioms identify a non-empty choice set and underline conflicts between chosen elements in strict preference circuits. Finally, we exploit the close link between this family and transitive closures to compute choice sets in linear time, followed by a relevant practical application.     

Spectra of digraphs

In this primarily expository paper we survey classical and some more recent results on the spectra of digraphs, equivalently, the spectra of (0,1)-matrices, with emphasis on the spectral radius.     

有向图字典乘积的代数群性质

字典乘积有向图G_1→⊙G_2是通过已知阶数较小的有向图G_1和G_2构造来的,这些小有向图G_1和G_2的拓扑结构和性质肯定影响大有向图G_1→⊙G_2的拓扑结构和性质.运用群论方法,证明了有向图字典乘积的一些代数性质,如:结合律、分配律等.     

On Schreier graphs of gyrogroup actions

The notion of a group action can be extended to the case of gyrogroups. In this article, we examine a digraph and graph associated with a gyrogroup action on a finite nonempty set, called a Schreier digraph and graph. We show that algebraic properties of gyrogroups and gyrogroup actions such as being gyrocommutative, being transitive, and being fixed-point-free are reflected in their Schreier digraphs and graphs. We also prove graph-theoretic versions of the three fundamental theorems involving actions: the Cauchy–Frobenius lemma (also known as the Burnside lemma), the orbit-stabilizer theorem, and the orbit decomposition theorem. Finally, we make a connection between gyrogroup actions and actions of symmetric groups by evaluation via Schreier digraphs and graphs.     

A new family of proximity graphs: Class cover catch digraphs

Motivated by issues in machine learning and statistical pattern classification, we investigate a class cover problem (CCP) with an associated family of directed graphs—class cover catch digraphs (CCCDs). CCCDs are a special case of catch digraphs. Solving the underlying CCP is equivalent to finding a smallest cardinality dominating set for the associated CCCD, which in turn provides regularization for statistical pattern classification. Some relevant properties of CCCDs are studied and a characterization of a family of CCCDs is given.     

Transitive digraphs with more than one end

In this paper we investigate infinite, locally finite, connected, transitive digraphs with more than one end. For undirected graphs with these properties it has been shown that they are trees as soon as they are 2-arc transitive. In the case of digraphs the situation is much more involved. We show that these graphs can have both thick and thin ends, even if they are highly arc transitive. Hence they are far away from being ‘tree-like’. On the other hand all known examples of digraphs with more than one end are either highly arc transitive or at most 1-arc transitive. We conjecture that infinite, locally finite, connected, 2-arc transitive digraphs with more than one end are highly arc transitive and prove that this conjecture holds for digraphs with prime in- and out-degree and connected cuts.     

Locally semicomplete digraphs: A generalization of tournaments

In this paper we introduce a new class of directed graphs called locally semicomplete digraphs. These are defined to be those digraphs for which the following holds: for every vertex x the vertices dominated by x induce a semicomplete digraph and the vertices that dominate x induce a semicomplete digraph. (A digraph is semicomplete if for any two distinct vertices u and ν, there is at least one arc between them.) This class contains the class of semicomplete digraphs, but is much more general. In fact, the class of underlying graphs of the locally semi-complete digraphs is precisely the class of proper circular-arc graphs (see [13], Theorem 3). We show that many of the classic theorems for tournaments have natural analogues for locally semicomplete digraphs. For example, every locally semicomplete digraph has a directed Hamiltonian path and every strong locally semicomplete digraph has a Hamiltonian cycle. We also consider connectivity properties, domination orientability, and algorithmic aspects of locally semicomplete digraphs. Some of the results on connectivity are new, even when restricted to semicomplete digraphs.     

Backwards genetic inheritance through coalgebra-graphs

We attach two weighted digraphs to any coalgebra C with genetic realization. Algebraically, these digraphs describe the right and left coideal structure of C, whereas genetically they represent the backwards inheritance through the male and female heritage lines, respectively. After establishing the relationship between the strongly connectedness of the attached digraphs and the simplicity of the genetic coalgebras, both digraphs are unified by considering the notion of genetic trigraph.     

Adjusted Interval Digraphs

Interval digraphs were introduced by West et al. They can be recognized in polynomial time and admit a characterization in terms of incidence matrices. Nevertheless, they do not have a forbidden structure characterization nor a low-degree polynomial recognition algorithm.We introduce a new class of ‘adjusted interval digraphs,’ obtained by a slight change in the definition. We show that, by contrast, these digraphs have a natural forbidden structure characterization, parallel to a characterization for undirected graphs, and admit a simple recognition algorithm. We relate adjusted interval digraphs to a list homomorphism problem. Each digraph H defines a corresponding list homomorphism problem L-HOM(H). We observe that if H is an adjusted interval digraph, then the problem L-HOM(H) is polynomial time solvable, and conjecture that for all other reflexive digraphs H the problem L-HOM(H) is NP-complete. We present some preliminary evidence for the conjecture, including a proof for the special case of semi-complete digraphs.     

Ihara zeta functions of digraphs

After defining and exploring some of the properties of Ihara zeta functions of digraphs, we improve upon Kotani and Sunada’s bounds on the poles of Ihara zeta functions of undirected graphs by considering digraphs whose adjacency matrices are directed edge matrices.     

On Vosperian and Superconnected Vertex-Transitive Digraphs

We investigate the structure of a digraph having a transitive automorphism group where every cutset of minimal cardinality consists of all successors or all predecessors of some vertex. We give a complete characterization of vosperian arc-transitive digraphs. It states that an arc-transitive strongly connected digraph is vosperian if and only if it is irreducible. In particular, this is the case if the degree is coprime with the order of the digraph. We give also a complete characterization of vosperian Cayley digraphs and a complete characterization of irreducible superconnected Cayley digraphs. These two last characterizations extend the corresponding ones in Abelian Cayley digraphs and the ones in the undirected case.     

Walkwise and admissible mappings between digraphs

We define two classes of mappings, between digraphs, which are closely related to homomorphisms and pathwise homomorphisms of finite automata. We show that the cycle rank cannot increase under mappings of these types, derive various decomposition properties, and then relate these mappings to homomorphisms of digraphs.