Regular Digraphs Research Articles

PageRank regular digraphs with prime out-degrees

The PageRank algorithm was designed to rank web pages according to their relevance in the web. In this context, web topology is modeled as a directed graph whose vertices correspond to web pages with arcs representing hyperlinks. The algorithm assigns a score to each vertex, representing the relevance of its corresponding page in the web. In this paper we define a PageRank regular digraph, as a digraph whose vertices have all the same PageRank score. Such digraphs are interesting in the scope of privacy preserving release of digraph data in environments where a dishonest analyst may have previous structural knowledge about the PageRank relevance of some vertices. In this paper we analyze some structural properties of PageRank regular digraphs and provide a characterization when vertices have prime out-degree.

Diffusion of competing innovations in influence networks

Although an influence network is a crucial factor affecting diffusion processes, it is generally fixed or assumed to follow some typical topologies in traditional diffusion research. The purpose of this study is to examine how innovation diffusion is changed by different influence relationship structures existing among individuals. We introduce an extended version of the regular digraph that can represent various influence relationships among influentials and followers. We focus on three key features of influence relationships (i.e. monopolization, localization, and diversification of opinions) and examine how they are associated with certain macroscopic behavioral regularities by employing agent-based simulation. The simulation results show that market becomes “locked-in” to a single product when influences are monopolized by few influentials. We also find that when influence relationship becomes complex as influentials and followers increase, market cannot be categorized by a single typology, but becomes random and unpredictable. Our model demonstrates successfully an underlying principle of collective behavior that uniformity of behavior is promoted under monopolization of opinions and random, unpredictable behavioral patterns emerge from diversification of opinions.

ON THE REGULAR DIGRAPH OF IDEALS OF COMMUTATIVE RINGS

AbstractLet$R$be a commutative ring. The regular digraph of ideals of$R$, denoted by$\Gamma (R)$, is a digraph whose vertex set is the set of all nontrivial ideals of$R$and, for every two distinct vertices$I$and$J$, there is an arc from$I$to$J$whenever$I$contains a nonzero divisor on$J$. In this paper, we study the connectedness of$\Gamma (R)$. We also completely characterise the diameter of this graph and determine the number of edges in$\Gamma (R)$, whenever$R$is a finite direct product of fields. Among other things, we prove that$R$has a finite number of ideals if and only if$\mathrm {N}_{\Gamma (R)}(I)$is finite, for all vertices$I$in$\Gamma (R)$, where$\mathrm {N}_{\Gamma (R)}(I)$is the set of all adjacent vertices to$I$in$\Gamma (R)$.

The regular digraph of ideals of a commutative ring

Let R be a commutative ring and Max (R )b e the set of max- imal ideals of R. The regular digraph of ideals of R, denoted by − − → Γreg(R), is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J, there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R ,d e- noted by Γreg(R), has an edge joining I and J whenever either I contains a J-regular element or J contains an I-regular element. Here, for every Artinian ring R ,w e prove that|Max (R) |− 1 ω(Γreg(R)) |Max (R)| and χ(Γreg(R)) =2 |Max (R) |− k − 1, where k is the number of fields, appeared in the decompo- sition of R to local rings. Among other results, we prove that − − → Γreg(R) is strongly connected if and only if R is an integral domain. Finally, the diameter and the girth of the regular graph of ideals of Artinian rings are determined.

Spectra of transformation digraphs of a regular digraph

Let D = (V(D), A(D)) be a digraph, x, y, z be three variables taking values − or +, and D xyz denote the transformation digraphs of D. In this article, we determine the spectra of seven transformation digraphs of regular digraph D in terms of the spectrum of D.

Openly disjoint circuits through a vertex in regular digraphs

In 1985, Thomassen [14] constructed for every positive integer r , finite digraphs D of minimum degree δ ( D ) = r which do not contain a vertex x lying on three openly disjoint circuits, i.e. circuits which have pairwise exactly x in common. In 2005, Seymour [11] posed the question, whether an r -regular digraph contains a vertex x such that there are r openly disjoint circuits through x . This is true for r ≤ 3 , but does not hold for r ≥ 8 . But perhaps, in contrast to the minimum degree, a high regularity degree suffices for the existence of a vertex lying on r openly disjoint circuits also for r ≥ 4 . After a survey of these problems, we will show that every r -regular digraph with r ≥ 7 has a vertex which lies on 4 openly disjoint circuits.

Directed Hamilton Cycle Decompositions of the Tensor Products of Symmetric Digraphs

In this paper, the existence of directed Hamilton cycle decompositions of symmetric digraphs of tensor products of regular graphs, namely, $$(K_r \times K_s)^*,\,\,((K_r \circ \overline{K}_s) \times K_n)^*,\,\,((K_r \times K_s) \times K_m)^*,\,\,((K_r \circ \overline{K}_s) \times (K_m \circ \overline{K}_n))^*$$and $$(K_{r,r} \times (K_m \circ \overline{K}_n))^*$$, where × and ∘ denote the tensor product and the wreath product of graphs, respectively, are proved. In [16], Ng has obtained a partial solution to the following conjecture of Baranyai and Szasz [6], see also Alspach et al. [1]: If D 1 and D 2 are directed Hamilton cycle decomposable digraphs, then D 1 ∘ D 2 is directed Hamilton cycle decomposable. Ng [17] also has proved that the complete symmetric r-partite regular digraph, $$K_{r(s)}^{*} = (K_r \circ \overline{K}_s)^*$$, is decomposable into directed Hamilton cycles if and only if $$(r,s) \ne (4,1)$$or (6, 1); using the results obtained here, we give a short proof of it, when $$r \notin {4,6}$$.

Domination in a digraph and in its reverse

Let D be a digraph. By γ ( D ) we denote the domintaion number of D and by D − we denote a digraph obtained by reversing all the arcs of D . In this paper we prove that for every δ ≥ 3 and k ≥ 1 there exists a simple strongly connected δ -regular digraph D δ , k such that γ ( D δ , k − ) − γ ( D δ , k ) = k . Analogous theorem is obtained for total domination number provided that δ ≥ 4 .

4-cycles in mixing digraphs

Abstract It is known that every simple graph with n 3 / 2 edges contains a 4-cycle. A similar statement for digraphs is not possible since no condition on the number of arcs can guarantee an (oriented) 4-cycle. We find a condition which does guarantee the presence of a 4-cycle and our result is tight. Our condition, which we call f-mixing, can be seen as a quasirandomness condition on the orientation of the digraph. We also investigate the notion of mixing for regular and almost regular digraphs. In particular we determine how mixing a random orientation of a random graph is.

Multidimensional Manhattan Street Networks

We formally define the $n$-dimensional Manhattan street network $M_n$—a special case of an $n$-regular digraph—and we study some of its structural properties. In particular, we show that $M_n$ is a Cayley digraph, which can be seen as a subgroup of the $n$-dimensional version of the wallpaper group $pgg$. These results induce a useful new representation of $M_n$, which can be applied to design a local (shortest-path) routing algorithm and to study some other metric properties, such as the diameter. We also show that the $n$-dimensional Manhattan street networks are Hamiltonian and, in the standard case (that is, in dimension two), we give sufficient conditions for a $2$-dimensional Manhattan street network to be decomposable into two arc-disjoint Hamiltonian cycles.

On the structure of the adjacency matrix of the line digraph of a regular digraph

We show that the adjacency matrix M of the line digraph of a d-regular digraph D on n vertices can be written as M = AB , where the matrix A is the Kronecker product of the all-ones matrix of dimension d with the identity matrix of dimension n and the matrix B is the direct sum of the adjacency matrices of the factors in a dicycle factorization of D.

Hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs

For a nonnegative n × n matrix A, we find that there is a polynomial f ( x ) ∈ R [ x ] such that f( A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f( x) with tr f( A) = n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n × n matrix A, we are led to define its Hoffman polynomial to be the polynomial f( x) of minimum degree satisfying that f( A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials.

Independence number of iterated line digraphs

In this paper, we deal with the independence number of iterated line digraphs of a regular digraph G. We give pertinent lower bounds and give an asymptotic estimation of the ratio of the number of vertices of a largest independent set of the n th iterated line digraph of G.

On Arc-Regular Permutation Groups Using Latin Squares

For a given a permutation group G, the problem of determining which regular digraphs admit G as an arc-regular group of automorphism is considered. Groups which admit such a representation can be characterized in terms of generating sets satisfying certain properties, and a procedure to manufacture such groups is presented. The technique is based on constructing appropriate factorizations of (smaller) regular line digraphs by means of Latin squares. Using this approach, all possible representations of transitive groups of degree up to seven as arc-regular groups of digraphs of some degree is presented.

Doubly regular digraphs and symmetric designs

If an incidence matrix N of a symmetric design is such that N+ N t is a (0,1) matrix, then N is an adjacency matrix of a doubly regular asymmetric digraph, and vice versa. We construct several parametrically new infinite families of such digraphs. To carry on some of these constructions, we obtain an infinite family of skew balanced generalized weighing matrices.

The underlying line digraph structure of some (0,1)-matrix equations

From the theory of Hoffman polynomial, it is known that the adjacency matrix A of a strongly connected regular digraph of order n satisfies certain polynomial equation A l P ( A )= J n , where l is a nonnegative integer, P ( x ) is a polynomial with rational coefficients, and J n is the n × n matrix of all ones. In this paper we present some sufficient conditions, in terms of the coefficients of P ( x ), to ensure that all (0,1)-matrices satisfying such an equation with l >0 have an underlying line digraph structure, that is to say, for any solution A there exists a (0,1)-matrix C satisfying P ( C )= J n / d l and the associated (d-regular) digraph of A, Γ ( A ), is the lth iterated line digraph of Γ ( C ). As a result, we simplify the study of some digraph classes with order functions asymptotically attaining the Moore bound.

TSP tour domination and Hamilton cycle decompositions of regular digraphs

In this paper, we solve a problem by Glover and Punnen (J. Oper. Res. Soc. 48 (1997) 502–510) from the context of domination analysis, where the performance of a heuristic algorithm is rated by the number of solutions that are not better than the solution found by the algorithm, rather than by the relative performance compared to the optimal value. In particular, we show that for the asymmetric traveling salesman problem, there is a deterministic polynomial time algorithm that finds a tour that is at least as good as the median of all tour values. Our algorithm uses an unpublished theorem by Häggkvist on the Hamilton decomposition of regular digraphs.

Construction of [formula omitted]-arc transitive digraphs

A digraph is k-arc transitive if it has a group of automorphisms which acts transitively on the set of k-arcs. Unlike the undirected case, in which the cycles are the only k-arc transitive finite graphs for k⩾8, there are k-arc transitive finite digraphs with arbitrary out-degree for every positive integer k. We show that every regular finite digraph admits a covering digraph which is k-arc transitive. The result provides a technique to construct k-arc transitive digraphs from arbitrary regular digraphs. Some examples are given from complete graphs with and without loops.

Cayley Digraphs Based on the de Bruijn Networks

A construction of Cayley digraphs associated to arc-colored regular digraphs is presented. The resulting Cayley digraphs, which we call Cayley regular covers, can be seen as a symmetrization of the original digraph. This construction is applied to the de Bruijn digraphs. By using the fact that they are iterated line digraphs of complete symmetric digraphs, valuable information about their Cayley regular covers regarding routings, diameter, hamiltonicity, fault-tolerance properties and degree of symmetry is obtained. In particular, a shortest-path, self-routing algorithm is given for a family of Cayley digraphs which includes the well known butterfly network. These results can be applied to the design of permutation networks. The Cayley regular covers represent sets of permutations in the original digraph which can be performed without conflict. In particular, a sharply 2-transitive group of permutations on the de Bruijn network is presented which admits a simple shortest-path self-routing algorithm. By using the same construction, a Cayley digraph on the symmetric group on the nodes of the de Bruijn digraph of degree two is obtained. The techniques introduced in this paper can also be extended to other families of iterated line digraphs.

The alternating polynomials and their relation with the spectra and conditional diameters of graphs

Given a graph Γ on n = ¦VΓ¦ vertices, the distance between two subgraphs Γ 1, Γ 2 ⊂ Γ, denoted by ∂( Γ 1, Γ 2), is the minimum among the distances between vertices of Γ 1 and Γ 2. For some integers 1 ⩽ s, t ⩽ n, the conditional ( s, t)- diameter of Γ is then defined as D (s,t) = max Γ 1, Γ 2 ⊂ Γ {∂(Γ 1, Γ 2): ¦VΓ 1¦ = s, ¦VΓ 2¦ = t} . Let Γ have distinct eigenvalues λ > λ 1 > λ 2 > … > λ d . For every k = 0, 1, …, d − 1, the k- alternating polynomial P k is defined to be the polynomial of degree k and norm ‖P k‖∞ = max 1⩽ / ⩽d {¦P k(λ l)¦} = 1 that attains maximum value at λ. These polynomials, which may be thought of as the discrete version of the Chebychev ones, were recently used by the authors to bound the (standard) diameter D ≡ D 1,1 of Γ in terms of its eigenvalues. In this work we derive similar results for conditional diameters. For instance, it is shown that P k(λ)> ‖ν‖ 2 S −1 ⇒ D s,s⩽k , where v is the (positive) eigenvector associated to λ, with minimum component 1. Similar results are given for locally regular digraphs by using the Laplacian spectrum. Some applications to the study of other parameters, such as the connectivity of Γ, are also discussed.