Non-oriented Graphs Research Articles

THE MEDIAN TREE

Mathematical models in which objects have a finite discrete structure are used to solve a wide range of practical problems. Such are, for example, many problems in auto-matic design, operation research, control theory, network planning, and computational geometry. Often, oriented or non-oriented graphs are considered as such models, which can adequately describe the structure of the problem of interest to us. Among these tasks, a certain place is occupied by tasks related to network issues of optimal placement of ware-houses, certain service points, emergency, medical services, etc. When studying such issues, it becomes necessary to identify, in a certain sense, the central objects in the networks un-der consideration, and in order to find them, the concept of the shortest path between these two given elements plays an essential role. A definition of the medial vertex of a graph is provided in the article – the vertex whose sum of distances to the others is minimal. The median of a non-oriented tree con-sisting of medial vertices is considered, and it is proved that the median of each such tree has either one vertex or two connected by an edge. This result is a definite generalization of a similar property of the center of a tree consisting of central vertices. The question of the relationship between the center and median of the same tree is also studied. The examples of trees with ten vertices show that such a direct connection does not exist, and there can be all four logically possible cases: the same tree can contain one or two central and one or two medial vertices.

Задача о беспорядках

The problem of characterizing symmetric necklaces as disorders is considered. Such an idea is an alternative approach to the one using Poya's enumeration theorem, including Burnside's lemma. The relationship between partitions of numbers and permutation types is shown. Disorder types are associated with necklace types considered as permutations in which no element remains in place, as well as with disorders with fixed elements. The distributions of disorders by types for necklaces up to the seventh degree are studied. The number and enumeration of disorders of each type is carried out, and their evenness is established. Particular attention is paid to the study of the symmetric necklaces properties. Classes of symmetry and asymmetry are enumerated for each necklace type considered. At the same time, the concepts of chiral and achiral symmetry are introduced as varieties of axial symmetry. Relationships between the power of symmetry classes and the order of symmetry of necklaces are revealed. The concept of symmetry diagrams for necklaces is introduced. Diagrams could be applied for determining the properties of invariance of symmetric and asymmetric necklaces of a given length. The concept of congruence is used as an equivalence relation between necklaces. It allows implementing a geometric approach to the study of necklaces and the visualization of the results obtained. For this purpose, necklaces are associated with multiconnected non-oriented graphs, the vertices of which are the vertices of regular polygons. In this case, the number of vertices in the graph corresponds to the necklace’s length, and the number of connected components coincides with the number of bead colors. As an application of the results obtained, the possibility of studying the accent dynamics of poetic stanzas vertically using necklaces is considered. Empirical data on the use of various types of seven-line stanzas in poetic practice are presented. It is noted that approximately a quarter of the existing symmetric types of necklaces for various reasons do not find practical application.

Ford-Fulkerson algorithm refinement for the cooperation effectiveness increase of intensive and low-density lines

The article describes tools of the railway control on intensive and low-density lines which is directed on the effectiveness increase of low-density line functioning, for the solution of perspective tasks of the railway network functioning and development. For the too realization the oriented graph with the Ford-Fulkerson algorithm which allows determining the maximum flow and the minimum cut for non-oriented graphs. Firstly as values of graph tops inverse values of the station rating and as graph edges inverse values of the railway line class are accepted. The use of this approach allows determining the maximum flow in the system and provides the clear view of relations of transportation capacities of railway lines and stations.

Orienting supersingular isogeny graphs

AbstractWe introduce a category of 𝓞-oriented supersingular elliptic curves and derive properties of the associated oriented and nonoriented ℓ-isogeny supersingular isogeny graphs. As an application we introduce an oriented supersingular isogeny Diffie-Hellman protocol (OSIDH), analogous to the supersingular isogeny Diffie-Hellman (SIDH) protocol and generalizing the commutative supersingular isogeny Diffie-Hellman (CSIDH) protocol.

Application of the Ford-Falkerson algorithm to detect excess information

The object of the article is the network maximum flow algorithm, mainly the Ford-Fulkerson algorithm. The algorithm began to be developed by two scientists: Ford and Fulkerson. This algorithm was proposed in order to find the maximum flow in the network. They began to be actively studied by scientists from the middle of the last century. The first report of "Maximum Network Flow" dates back to 1954. The authors of the report, Ford and Fulkerson had proved the theorem on the maximum flow and the minimum cross section for non-oriented graphs: the value of the maximum flow in the network is equal to the minimum throughput capacity of the section. The interest in the solution of these tasks was primarily due to practical needs, for that time construction of routes for the transportation of raw materials was not optimal and transported more raw materials than can transfer the connection between points. Such problems often arise when constructing connections that transport oil through pipes or transport coal through special excavators. The subject of the article is the problem of finding the maximum flow in the network. In graph theory, the transport network is an indicative graph in which each arc has no negative throughput and flow. Two peaks are distinguished: source and drain - such that any other vertex in the network lies on the way from source to drain. The article consists of two sections. In the first section we consider the mathematical formulation of the problem and concrete examples of problems. The second section examines the classic Ford-Fulkerson algorithm, the modified Ford-Fulkerson algorithm to find excess information on the network, and the work of a modified algorithm on a specific example from the first section. The considered problems are relevant both from a theoretical and a practical point of view.

Formulation of the Problem of Maximum Clique Determination in Non-Oriented Graphs

In the train traffic organization, large amount of information should be processed in real time. Thereby in complicated dispatching systems, rational use of resources is needed. To perform this work a model of a management system is constructed and it is represented as a sparse graph. Further optimization of the model requires solving the Maximum Clique Problem (MCP) for the less time than the exponential time. This article contains two procedures for solving the task for subexponential time. Procedure B accurately estimates the size of the maximum clique in the graph and performs the sorting of the vertices, in such a way that the vertices that are not exactly in the maximum clique can be rejected. Procedure A, in fact, finds cliques of the maximum size in a given graph, using the procedure B. The main advantage of this method is that using it is expedient in real time for sufficiently large graphs, which in turn is important for the construction of control systems.

Poisson Brackets Symmetry from the Pentagon-Wheel Cocycle in the Graph Complex

Kontsevich designed a scheme to generate infinitesimal symmetries $\dot{\mathcal{P}} = \mathcal{Q}(\mathcal{P})$ of Poisson brackets $\mathcal{P}$ on all affine manifolds $M^r$; every such deformation is encoded by oriented graphs on $n+2$ vertices and $2n$ edges. In particular, these symmetries can be obtained by orienting sums of non-oriented graphs $\gamma$ on $n$ vertices and $2n-2$ edges. The bi-vector flow $\dot{\mathcal{P}} = \text{Or}(\gamma)(\mathcal{P})$ preserves the space of Poisson structures if $\gamma$ is a cocycle with respect to the vertex-expanding differential in the graph complex. A class of such cocycles $\boldsymbol{\gamma}_{2\ell+1}$ is known to exist: marked by $\ell \in \mathbb{N}$, each of them contains a $(2\ell+1)$-gon wheel with a nonzero coefficient. At $\ell=1$ the tetrahedron $\boldsymbol{\gamma}_3$ itself is a cocycle; at $\ell=2$ the Kontsevich--Willwacher pentagon-wheel cocycle $\boldsymbol{\gamma}_5$ consists of two graphs. We reconstruct the symmetry $\mathcal{Q}_5(\mathcal{P}) = \text{Or}(\boldsymbol{\gamma}_5)(\mathcal{P})$ and verify that $\mathcal{Q}_5$ is a Poisson cocycle indeed: $[\![\mathcal{P},\mathcal{Q}_5(\mathcal{P})]\!]\doteq 0$ via $[\![\mathcal{P},\mathcal{P}]\!]=0$.

On switching classes of graphs

A class of non-oriented simple graphs is called Seidel switching self-complementary (s.s.c. for short) if the complement of any representing graph is in the same equivalence class. Relating the Seidel adjacency matrix of a graph with a Gram matrix, we introduce the 3-signature (s,t) of a switching class of n-vertex graphs. The numbers s and t are the numbers of positive and negative triples within any representing graph of the class. It appears that, for any s.s.c. class of n-vertex graphs, these numbers are equal, yielding (n3) even. Consequently if n≡3(mod4) then there is no s.s.c. class of n-vertex graphs. We also prove that this 3-signature depends only on the spectrum of the adjacency matrix of the graph. We then consider the switching classes of Paley conference graphs with 4k+2 vertices, 4k+1=pα, p an odd prime and α a positive integer. We reprove that these classes are s.s.c. Moreover, it is proven that all 4k-vertex graphs contained in a (4k+2)-vertex Paley conference graph are switching equivalent and their class is still a s.s.c. class. In addition, the 3-signature is generalized in view of obtaining a complete invariant of switching classes up to order 8.

Covering and Euler cycles on non-oriented graphs

A covering cycle is a closed path that traverses each edge of a graph at least once. Two cycles are equivalent if one is a cyclic permutation of the other. We compute the number of equivalence classes of non-periodic covering cycles of given length in a non-oriented connected graph. A special case is the number of Euler cycles (covering cycles that cover each edge of the graph exactly once) in the non-oriented graph. We obtain an identity relating the numbers of covering cycles of any length in a graph to a product of determinants.

Метод перечисления максимальных независимых множеств в неориентированных графах

Метод перечисления максимальных независимых множеств в неориентированных графах

The heptagon-wheel cocycle in the Kontsevich graph complex

The real vector space of non-oriented graphs is known to carry a differential graded Lie algebra structure. Cocycles in the Kontsevich graph complex, expressed using formal sums of graphs on n vertices and 2n — 2 edges, induce - under the orientation mapping - infinitesimal symmetries of classical Poisson structures on arbitrary finite-dimensional affine real manifolds. Willwacher has stated the existence of a nontrivial cocycle that contains the (2ℓ+1)-wheel graph with a nonzero coefficient at every ℓ∈ℕ. We present detailed calculations of the differential of graphs; for the tetrahedron and pentagon-wheel cocycles, consisting at ℓ = 1 and ℓ = 2 of one and two graphs respectively, the cocycle condition d(γ) = 0 is verified by hand. For the next, heptagon-wheel cocycle (known to exist at ℓ = 3), we provide an explicit representative: it consists of 46 graphs on 8 vertices and 14 edges.

Paths on Ordered Edges in Non-oriented Graphs and Economic Networks Modeling

Paths on Ordered Edges in Non-oriented Graphs and Economic Networks Modeling

The minimum reload [formula omitted]– [formula omitted] path, trail and walk problems

This paper deals with problems on non-oriented edge-colored graphs. The aim is to find a route between two given vertices s and t . This route can be a walk, a trail or a path. Each time a vertex is crossed by a walk there is an associated non-negative reload cost r i , j , where i and j denote, respectively, the colors of successive edges in this walk. The goal is to find a route whose total reload cost is minimum. Polynomial algorithms and proofs of NP-hardness are given for particular cases: when the triangle inequality is satisfied or not, when reload costs are symmetric (i.e., r i , j = r j , i ) or asymmetric. We also investigate bounded degree graphs and planar graphs. We conclude the paper with the traveling salesman problem with reload costs.

THREE-PAGE ENCODING AND COMPLEXITY THEORY FOR SPATIAL GRAPHS

A finitely presented semigroup RSGnis constructed for n ≥ 2. The centre of RSGnencodes uniquely up to rigid ambient isotopy in 3-space all nonoriented spatial graphs with vertices of degree ≤ n. This encoding is obtained by using three-page embeddings of graphs into the three-page book T × I, where T is the cone on three points, and I is the unit segment. The notion of the three-page complexity for spatial graphs is introduced via three-page embeddings. This complexity satisfies the properties of finiteness and additivity under natural operations.

Kinematic analysis of bevel-gear trains using graphs

The non-oriented and oriented graph techniques are used for kinematic analysis of bevel-gear trains. In both techniques, the kinematic structure of the gear trains is represented by a graph. Although the non-oriented graphs are simple to draw, they are used only to determine the carrier nodes of the fundamental circuits. On the other hand, the oriented graphs carry more information than the non-oriented graphs since each line in an oriented graph represents a pair of complementary variables. In this paper, these two techniques are compared and the advantages of the oriented graph technique are demonstrated by the kinematic analysis of an articulated robotic mechanism used by the Cincinnati Milacron T3.

Chevalley cohomology for Kontsevich’s graphs

We introduce the Chevalley cohomology for the graded Lie algebra of polyvector ?elds on Rd. This cohomology occurs naturally in the problem of construction and classi?cation of formalities on the space Rd. Considering only graph formalities, that is, formalities de?ned with the help of graphs as in the original construction of Kontsevich, we de?ne (as the ?rst and third authors did earlier for the Hochschild cohomology) the Chevalley cohomology directly on spaces of graphs. More precisely, observing ?rst a noteworthy property for Kontsevich?s explicit formality on Rd, we restrict ourselves to graph formalities with that property. With this restriction, we obtain some simple expressions for the Chevalley coboundary operator; in particular, we can write this cohomology directly on the space of purely aerial, nonoriented graphs. We also give examples and applications.

Variations sur le thème [formula omitted

Let G=( X, Y; E) be a bipartite graph with bipartite complement G=(X,Y; E) . The number of r-matchings of G is denoted by p( G, r). It is classical that the vector [p( G,r)] r=1,2,… is determined by [ p( G, r)] r=1,2,… . We make this statement more explicit by proving new duality theorems, generalizing and globalizing the results of Chow, Foata, Gessel, Joni, Rota and Zeilberger, in particular. For oriented graphs this provides a short proof of Berge's fundamental identity between Hamiltonian paths and circuits. Expressed in terms of set functions, the identity immediately implies the Chung–Graham conjecture (first derived by Chow and Gessel) as well as Rédei's, Lovász', and Camion's and Rao's parity results for tournaments, non-oriented and self-complementary graphs, respectively. Finally, we study the relations between set functions and symmetric functions and show that the main theorem in Chow's PhD thesis becomes a direct consequence of Berge's identity.

An artificial neural network approach for routing in distributed computer networks

This paper presents a new approach based on the Hopfield model of artificial neural networks to solve the routing problem in a context of computer network design. The computer networks considered are packet switching networks, modeled as non-oriented graphs where nodes represent servers, hosts or switches, while bi-directional and symmetric arcs represent full duplex communication links. The proposed method is based on a network representation enabling to match each network configuration with a Hopfield neural network in order to find the best path between any node pair by minimizing an energy function. The results show that the time delay derived from flow assignment carried out by this approach is, in most cases, better than those determined using conventional routing heuristics. Therefore, this neural-network approach is suitable to be integrated into an overall topological design process of moderate-speed and high-speed networks subject to quality of service constraints as well as to changes in configuration and link costs.

Dynnikov Three-Page Diagrams of Spatial 3-Valent Graphs

Recently, Dynnikov suggested a new way of encoding nonoriented links by three-page diagrams [1], and this permitted him to obtain an algebraic classification of the isotopy classes of nonoriented links (see [2]). We generalize this approach to 3-valent graphs. A finite graph is said to be 3-valent if exactly three edges meet at any vertex. We consider only 3-valent nonoriented graphs, which can be disconnected and have loops and multi-edges. By a spatial graph we mean an embedding of a graph in R3 under which the edges become finite polygonal lines. We study the spatial graphs up to an ambient isotopy, where an ambient isotopy between two graphs is a continuous family of homeomorphisms φt : R3 → R3 , t ∈ [0, 1], such that φ0 = id and φ1 sends one of these graphs to the other. Isotopy invariants of spatial graphs were studied in [3, 4]. For other equivalence relations (including concordance, homotopy, and homology), see [5]. The notion of spatial graph is motivated both theoretically and practically. First, the problem to classify the spatial graphs up to isotopy is a special case of the general topological problem of classifying the embeddings in Euclidean space. We simultaneously obtain a natural extension of the classical knot theory to more complex one-dimensional objects. Many invariants of ordinary links, including the Alexander polynomials and Vassiliev invariants, can be generalized to graphs [6, 7]. As well as the ordinary links, the spatial graphs can readily be represented by plane diagrams determined up to the following Reidemeister moves:

A Heuristic Algorithm of Recognition of Isomorphism of Graphs

A heuristic polynomial algorithm is presented, which is used for the recognition of isomorphism of graphs and can be assigned to the group of methods that use local characteristic invariants of graphs. At each step, the behavior of the algorithm depends on information obtained at its previous steps. All the theorems stated are proved for a class of nonoriented graphs.