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A digraph group is a group defined by non-empty presentation with the property that each relator is of the form $R(x, y)$ , where x and y are distinct generators and $R(\\cdot , \\cdot )$ is determined by some fixed cyclically reduced word $R(a, b)$ that involves both a and b. Associated with each such presentation is a digraph whose vertices correspond to the generators and whose arcs correspond to the relators. In this article, we consider digraph groups for strong digraphs that are digon-free and triangle-free. We classify when the digraph group is finite and show that in these cases it is cyclic, giving its order. We apply this result to the Cayley digraph of the generalized quaternion group, to circulant digraphs, and to Cartesian and direct products of strong digraphs.

The primary objective of this paper, is to introduce eight types of topologies on a finite digraphs and state the implication between these topologies. Also we used supra open digraphs to introduce a new types for approximation rough digraphs.

Chordal graphs are important in structural graph theory. Chordal digraphs are a digraph analogue of chordal graphs and have been a subject of active studies recently. Unlike chordal graphs, chordal digraphs lack many structural properties such as forbidden subdigraph or representation characterizations. In this paper we introduce the notion of semi-strict chordal digraphs which form a class strictly between chordal digraphs and chordal graphs. Semi-strict chordal digraphs have rich structural properties. We characterize semi-strict chordal digraphs in terms of knotting graphs, a notion analogous to the one introduced by Gallai for the study of comparability graphs. We also give forbidden subdigraph characterizations of semi-strict chordal digraphs within the classes of locally semicomplete digraphs and weakly quasi-transitive digraphs.

Endowing the set of functional graphs (FGs) with the sum (disjoint union of graphs) and product (standard direct product on graphs) operations induces on FGs a structure of a commutative semiring R. The operations on R can be naturally extended to the set of univariate polynomials R[X] over R. This paper provides a polynomial time algorithm for deciding if equations of the type AX=B have solutions when A is just a single cycle and B a set of cycles of identical size. We also prove a similar complexity result for some variants of the previous equation. Final version accepted by DMTCS; added a linefeed before 'Clearly' in the before last page, as asked by the editor

Abstract-A digraph that has arcs of two colours is called a two-coloured digraph. In this case, the colours used are red and black. Let d and k be non-negative integers, where d represents the number of red arcs and k represents the number of black arcs. A (d, k)-walk on the two-coloured digraph is defined as a walk with d red arcs and k black arcs. The smallest integer sum of d and k such that there is a (d, k)-walk from vertex y to vertex z is called the exponent number of two-coloured digraph, whereas the smallest integer sum of d and k such that there is (d, k)-walk from each vertex to vertex vx is called the inner local exponent of a vertex vx. This article discusses the inner local exponent of a two-cycle non-Hamiltonian twocoloured digraph with cycle lengths n and 3n +1. This digraph has exactly four red arcs. The four red arcs are combined consecutively or alternately when there is one allied vertex.

MSP DAGs (short for minimal series-parallel digraphs) can be defined from the single vertex graph by applying the parallel composition and series composition. We prove an upper bound of 6 for the directed clique-width of MSP DAGs and show how a directed clique-width 6-expression can be found in linear time. Our 6-expression can be used to construct an MSP DAG G from its binary decomposition tree T(G) in linear time. We apply our bound on the directed clique-width to conclude a number of algorithmic consequences for MSP DAGs.

Let D be a finite and simple digraph with vertex set V ( D ). The minimum degree δ of a digraph D is defined as the minimum value of its out-degrees and its in-degrees. If D is a digraph with minimum degree δ and edge-connectivity λ, then λ ≤ δ . A digraph is maximally edge-connected if λ = δ . A digraph is called super-edge-connected if every minimum edge-cut consists of edges incident to or from a vertex of minimum degree. In this note we show that a digraph is maximally edge-connected or super-edge-connected if the number of arcs is large enough.

Let 𝑆𝑛 be the class of bicyclic signed digraphs with 𝑛 vertices whose two signed directed even cycles are vertex-disjoint. In this paper, we characterize the ordering of bicyclic signed digraphs in 𝑆𝑛 by energy with two positive or negative directed even cycles (resp., one positive directed even cycle and one negative directed even cycle). Furthermore, we determine extremal energy in 𝑆𝑛 by the two orderings.

This paper deals with the design and analysis of dynamical systems on directed graphs (digraphs) that achieve weight-balanced and doubly stochastic assignments. Weight-balanced and doubly stochastic digraphs are two classes of digraphs that play an essential role in a variety of coordination problems, including formation control, agreement, and distributed optimization. We refer to a digraph as doubly stochasticable (weight-balanceable) if it admits a doubly stochastic (weight-balanced) adjacency matrix. This paper studies the characterization of both classes of digraphs, and introduces distributed dynamical systems to compute the appropriate set of weights in each case. It is known that semiconnectedness is a necessary and sufficient condition for a digraph to be weight-balanceable. The first main contribution is a characterization of doubly-stochasticable digraphs. As a by-product, we unveil the connection of this class of digraphs with weight-balanceable digraphs. The second main contribution is the synthesis of a distributed strategy running synchronously on a directed communication network that allows individual agents to balance their in- and out-degrees. We show that a variation of our distributed procedure over the mirror graph has a much smaller time complexity than the currently available centralized algorithm based on the computation of the graph cycles. The final main contribution is the design of two cooperative strategies for finding a doubly stochastic weight assignment. One algorithm works under the assumption that individual agents are allowed to add self-loops. For the case when this assumption does not hold, we introduce an algorithm distributed over the mirror digraph which allows the agents to compute a doubly stochastic weight assignment if the digraph is doubly stochasticable and announce otherwise if it is not. Various examples illustrate the results.

Acyclic digraphs are the underlying representation of Bayesian networks, a widely used class of probabilistic graphical models. Learning the underlying graph from data is a way of gaining insights about the structural properties of a domain. Structure learning forms one of the inference challenges of statistical graphical models. Markov chain Monte Carlo (MCMC) methods, notably structure MCMC, to sample graphs from the posterior distribution given the data are probably the only viable option for Bayesian model averaging. Score modularity and restrictions on the number of parents of each node allow the graphs to be grouped into larger collections, which can be scored as a whole to improve the chain's convergence. Current examples of algorithms taking advantage of grouping are the biased order MCMC, which acts on the alternative space of permuted triangular matrices, and nonergodic edge reversal moves. Here, we propose a novel algorithm, which employs the underlying combinatorial structure of DAGs to define a new grouping. As a result convergence is improved compared to structure MCMC, while still retaining the property of producing an unbiased sample. Finally, the method can be combined with edge reversal moves to improve the sampler further. Supplementary materials for this article are available online.