Theory Of Digraphs Research Articles

Singular cell homology

In the paper we generalize the singular cubical homology theory of digraphs and introduce a collection of cell singular homologies that are parametrized by a natural parameter of the category of digraphs. The natural parameter corresponds to the “length” of a directed line digraph that corresponds to the unit segment in algebraic topology. The introduced earlier singular cubical homology is a particular case of this collection that corresponds to the line digraph with one arrow. We describe relations between these theories for different parameters given by a natural number and study their basic properties. In particular, we prove the functoriality and homotopy invariance of these theories and compare the theory with the path homology theory that was introduced in our previous papers. Then we describe the transfer of the cell singular homology theories to the categories of graphs, quivers, and multigraphs.

Magnitude homology and path homology

In this article, we show that magnitude homology and path homology are closely related, and we give some applications. We define differentials MH k ℓ ( G ) ⟶ MH k − 1 ℓ − 1 ( G ) $\operatorname{MH}^{\ell }_k(G) \longrightarrow \operatorname{MH}^{\ell -1}_{k-1}(G)$ between magnitude homologies of a digraph G $G$ , which make them chain complexes. Then we show that its homology MH k ℓ ( G ) $\mathcal {MH}^{\ell }_k(G)$ is non-trivial and homotopy invariant in the context of ‘homotopy theory of digraphs’ developed by Grigor'yan–Muranov–S.-T. Yau et al. (G-M-Ys in the following). It is remarkable that the diagonal part of our homology MH k k ( G ) $\mathcal {MH}^{k}_k(G)$ is isomorphic to the reduced path homology H ∼ k ( G ) $\tilde{H}_k(G)$ also introduced by G-M-Ys. Further, we construct a spectral sequence whose first page is isomorphic to magnitude homology MH k ℓ ( G ) $\operatorname{MH}^{\ell }_k(G)$ , and the second page is isomorphic to our homology MH k ℓ ( G ) $\mathcal {MH}^{\ell }_k(G)$ . As an application, we show that the diagonality of magnitude homology implies triviality of reduced path homology. We also show that H ∼ k ( g ) = 0 $\tilde{H}_k(g) = 0$ for k ⩾ 2 $k \geqslant 2$ and H ∼ 1 ( g ) ≠ 0 $\tilde{H}_1(g) \ne 0$ if any edges of an undirected graph g $g$ is contained in a cycle of length ⩾ 5 $\geqslant 5$ .

Advances in path homology theory of digraphs

Advances in path homology theory of digraphs

New Applied Problems in the Theory of Acyclic Digraphs

The following two optimization problems on acyclic digraph analysis are solved. The first of them consists of determining the minimum (in terms of volume) set of arcs, the removal of which from an acyclic digraph breaks all paths passing through a subset of its vertices. The second problem is to determine the smallest set of arcs, the introduction of which into an acyclic digraph turns it into a strongly connected one. The first problem was solved by reduction to the problem of the maximum flow and the minimum section. The second challenge was solved by calculating the minimum number of input arcs and determining the smallest set of input arcs in terms of the minimum arc coverage of an acyclic digraph. The solution of these problems extends to an arbitrary digraph by isolating the components of cyclic equivalence in it and the arcs between them.

The domination number of round digraphs

Abstract The concept of the domination number plays an important role in both theory and applications of digraphs. Let D = ( V , A ) D=(V,A) be a digraph. A vertex subset T ⊆ V ( D ) T\subseteq V(D) is called a dominating set of D, if there is a vertex t ∈ T t\in T such that t v ∈ A ( D ) tv\in A(D) for every vertex v ∈ V ( D ) \ T v\in V(D)\backslash T . The dominating number of D is the cardinality of a smallest dominating set of D, denoted by γ ( D ) \gamma (D) . In this paper, the domination number of round digraphs is characterized completely.

Homology of path complexes and hypergraphs

The path complex and its homology were defined in previous papers of the authors. The notion of a path complex is a natural discrete generalization of the notion of a simplicial complex. The theory of path complexes contains homotopy invariant homology theory of digraphs and (nondirected) graphs.In the paper we study the homology theory of path complexes. In particular, we describe functorial properties of paths complexes, introduce the notion of homotopy for path complexes and prove the homotopy invariance of path homology groups. We prove also several theorems that are similar to the results of classical homology theory of simplicial complexes. Then we apply this approach to construct homology theories on various categories of hypergraphs. We describe basic properties of these homology theories and relations between them. As a particular case, these results give new homology theories on the category of simplicial complexes.

On the path homology theory of digraphs and Eilenberg–Steenrod axioms

On the path homology theory of digraphs and Eilenberg–Steenrod axioms

Further restrictions on the structure of finite DCI-groups: an addendum

A finite group R is a $${\mathrm {DCI}}$$DCI-group if, whenever S and T are subsets of R with the Cayley digraphs $${\mathrm {Cay}}(R,S)$$Cay(R,S) and $${\mathrm {Cay}}(R,T)$$Cay(R,T) isomorphic, there exists an automorphism $$\varphi $$? of R with $$S^\varphi =T$$S?=T. The classification of $${\mathrm {DCI}}$$DCI-groups is an open problem in the theory of Cayley digraphs and is closely related to the isomorphism problem for digraphs. This paper is a contribution toward this classification, as we show that every dihedral group of order 6p, with $$p\ge 5$$p?5 prime, is a $${\mathrm {DCI}}$$DCI-group. This corrects and completes the proof of Li et al. (J Algebr Comb 26:161---181, 2007, Theorem 1.1) as observed by the reviewer (Conder in Math review MR2335710).

Idiosynchromatic Poetry

We prove equivalences of asymmetric partition relations involving natural numbers and products of weakly compact cardinals ?, infinite cardinals ?<? and natural numbers to certain classes of finitary problems in the theory of edge-coloured digraphs. We are able to determine three classes of Ramsey numbers exactly, typical examples in these classes are r(? 22,3), r(??2,3) and r(??3,3). We moreover provide general upper bounds for r(? 2 m,3) and r(??m,n).

Analysis of connected seismic events using methods of the theory of digraphs

For samples from the earthquakes catalogues with large seismic events the components of unilateral relatedness of corresponding orgraphs are studied. Definition of new mathematical adjectives is proposed — octopusindex of orgraph vertexes. Dynamics of this value changes can be used in algorithms of mediumterm forecast. Test examples and results of test calculations for the earthquakes of North California and the CrimeanBlack Sea region have been given.

The tractability index of memristive circuits: branch‐oriented and tree‐based models

The memory‐resistor or memristor is a new electrical element characterized by a nonlinear charge‐flux relation. This device poses many challenging problems, in particular from the circuit modeling point of view. In this paper, we address the index analysis of certain differential‐algebraic models of memristive circuits; specifically, our attention is focused on so‐called branch‐oriented models, which include in particular tree‐based formulations of the circuit equations. Our approach combines results coming from differential‐algebraic equation theory, matrix analysis and theory of digraphs. This framework should be useful in future studies of dynamical aspects of memristive circuits. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Improved generalized cell mapping for global analysis of dynamical systems

Three main parts of generalized cell mapping are improved for global analysis. A simple method, which is not based on the theory of digraphs, is presented to locate complete self-cycling sets that correspond to attractors and unstable invariant sets involving saddle, unstable periodic orbit and chaotic saddle. Refinement for complete self-cycling sets is developed to locate attractors and unstable invariant sets with high degree of accuracy, which can start with a coarse cell structure. A nonuniformly interior-and-boundary sampling technique is used to make the refinement robust. For homeomorphic dissipative dynamical systems, a controlled boundary sampling technique is presented to make generalized cell mapping method with refinement extremely accurate to obtain invariant sets. Recursive laws of group absorption probability and expected absorption time are introduced into generalized cell mapping, and then an optimal order for quantitative analysis of transient cells is established, which leads to the minimal computational work. The improved method is applied to four examples to show its effectiveness in global analysis of dynamical systems.

Potential of Fraser-Hipel-Kilgur Technique and Game Theory for Balkan conflicts: Pt. 1 Modelling Theory

In this paper, the potential applications of Fraser-Hipel-Kilgur methodology for conflict modelling and resolution, based on the theory of non-cooperative games theory, to the modelling analysis of Balkan conflicts is exploerd through the case study on the conflicting interests over R. Macedonia by the end of the twentieth century. Particular attention was devoted to the use of its advanced variant based on theory of digraphs and adjacent reachability matrices. The graph form takes game outcomes rather than individual decisions as the basic units for modelling conflicts, described in sufficient detail. A number of solution concepts can be formulated and employed for both two-player and multi-player games, the latter being of primary interst in here. The taxonomy of graph based solution concepts of the original methodology has been employed to account for certain anticipation horizon by each of the players. This methodology makes essential use of the fundamental axioms of game theory defined by posulates of the individual rationality, the players inertia, the limited-move stability, and the minimum effect for the opponents.

Potential of Fraser-Hipel-Kilgur Technique and Game Theory for Balkan Conflicts: PT.2 Applications

Abstract In this paper, the potential applications of Fraser-Hipel methodology for conflict modelling and resolution, based on the theory of non-cooperative games theory, to the modelling analysis of Balkan conflicts is exploerd through the case study on the conflicting interests over R. Macedonia byl the end of the twentieth century. Particular attention was devoted to the use of its advanced variant based on theory of digraphs and adjacent reachability matrices. The graph form takes game outcomes rather than individual decisions as the basic units for modelling conflicts, described in sufficient detail. A number of solution concepts can be formulated and employed for both two-player and multi-player games, the latter being of primary interst in here. The taxonomy of graph based solution concepts of the original methodology has been employed to account for certain anticipation horizon by each of the players. This methodology makes essential use of the fundamental axioms of game theory defined by posulates of the individual rationality, the players inertia, the limited-move stability, and the minimum effect for the opponents.

GLOBAL ANALYSIS OF DYNAMICAL SYSTEMS USING POSETS AND DIGRAPHS

In this paper the resources of the theory of partially ordered sets (posets) and the theory of digraphs are used to aid the task of global analysis of nonlinear dynamical systems. The basic idea underpinning this approach is the primitive notion that a dynamical systems is simply an ordering machine which assigns fore-and-after relations for pairs of states. In order to make the linkage between the theory of posets and digraphs and dynamical systems, cell mapping is used to put dynamical systems in their discretized form and an essential concept of self-cycling sets is used. After a discussion of the basic notion of ordering, appropriate results from the theory of posets and digraphs are adapted for the purpose of determining the global evolution properties of dynamical systems. In terms of posets, evolution processes and strange attractors can be studied in a new light. It is believed that this approach offers us a new way to examine the multifaceted complex behavior of nonlinear systems. Computation algorithms are also discussed and an example of application is included.

New Topological Formula and Rapid Algorithm for Reliability Analysis of Complex Networks

Part I derives a new topological formula for the terminalpair reliability of complex networks. The formula generates only non-cancelling terms. The non-cancelling terms in the reliability expression correspond one-to-one with the acyclic subgraphs of the given probabilistic graph. Part II introduces the concept of neutral sequences in acyclic graphs; several of their important properties are established. Based on these results a powerful algorithm for generating the reliability expression is presented. The reliability expression is obtained in symbolic factored form. Examples indicate that the present algorithm is appreciably faster than earlier methods. The properties of cyclic and acyclic graphs established in this paper are significant new results in the theory of digraphs and have further ramifications and wider application than in reliability.

Cardinality of finite topologies

Let S denote a finite set of cardinal n. The discrete topology on S contains 2 n open sets; the indiscrete topology contains 2 open sets. A partial answer is given to the question: For which intermediate integers m is there a topology on S with cardinal m? It is shown that no topology, other than the discrete, has cardinal greater than 3/4 2 n . Other bounds are derived on the cardinality of connected, non- T 0, connected and non- T 0, and non-connected topologies. Proofs involve results in the theory of transitive digraphs.

Homotopy Theory for Digraphs

We introduce a homotopy theory of digraphs (directed graphs) and prove its basic properties, including the relations to the homology theory of digraphs constructed by the authors in previous papers. In particular, we prove the homotopy invariance of homologies of digraphs and the relation between the fundamental group of the digraph and its first homology group. The category of (undirected) graphs can be identified by a natural way with a full subcategory of digraphs. Thus we obtain also consistent homology and homotopy theories for graphs. Note that the homotopy theory for graphs coincides with the one constructed in [1] and [2].