Odd Number Of Nodes Research Articles

Generalisation of a synchronous solution of the parity problem on cyclic configurations over a non-circulant graph

The parity problem is a classical binary benchmark for addressing the computational ability and limitations of automata networks. It refers to conceiving a local rule to allow deciding whether the number of 1-states in the nodes of an arbitrary network is an odd or even number, without global access to the nodes. In its standard formulation, the automata network has an odd number of nodes whose states, arranged as a cyclic configuration, should converge to a fixed point of all 0s, if the initial configuration has an even number of 1s, or to a fixed point of all 1s, otherwise. It has been shown that a local rule alone is able to solve the problem in this formulation, including a synchronous solution we have recently shown, totally based on the local parity rule of the cellular automata elementary space (number 150), with a certain pattern of connection between the nodes. Here, we generalise this solution, showing how to construct many others, combining rule 150 with rules 170 and 240, which are the local shifts of the same space, in such a way that the original solution is just one among countless possibilities. Such solutions may have different convergence times for specific configurations, but are equivalent in the context of all configurations of a given size; also, various solutions form symmetric pairs, in terms of the actual rules they used. The solutions were evaluated computationally and presented here without their formal proofs, but empirical evidences strongly suggest that they can be obtained by the same kind of technique we used in the solution exclusively with rule 150.

Binary quadratic optimization problems that are difficult to solve by conic relaxations

We study conic relaxations including semidefinite programming (SDP) relaxations and doubly nonnegative programming (DNN) relaxations to find the optimal values of binary QOPs. The main focus of the study is on how the relaxations perform with respect to the rank of the coefficient matrix in the objective of a binary QOP. More precisely, for a class of binary QOP instances, which include the max-cut problem of a graph with an odd number of nodes and equal weight, we show numerically that (1) neither the standard DNN relaxation nor the DNN relaxation derived from the completely positive formulation by Burer performs better than the standard SDP relaxation, and (2) Lasserre’s hierarchy of SDP relaxations requires solving the SDP with the relaxation order at least ⌈n/2⌉ to attain the optimal value. The bound ⌈n/2⌉ for the max-cut problem of a graph with equal weight is consistent with Laurent’s conjecture in 2003, which was proved recently by Fawzi, Saunderson and Parrilo in 2015.

ALL-OPTICAL NETWORK DESIGN WITH EVEN AND ODD NODES

Abstract We have studied the effects of OLTs and OADMs in WDM optical networks. All optical networks have proved to be cost efficient and power saving in comparison to O-E-O networks. Cost of a network can further be reduced by minimizing the number of IP router ports and the number of wavelengths required. It has been already studied the number of IP router ports required per node and number of wavelengths required to carry a fixed amount of traffic, considering the network containing even and odd number of routing nodes. And finally the result has been compared with other architectures like point to point WDM and hub networks, finally all-optical networks proved to be most cost efficient in saving number of wavelength requirements and IP router port requirements. In this paper we have compared all-optical network with itself, by taking even and odd number of nodes. That is we have compared all-optical network containing even number of nodes with the same all-optical network containing odd number of nodes. The result what we obtained is to honor same amount of traffic “t”, all-optical network containing odd number of nodes require lesser number of wavelengths than its previous even number of nodes. We have here varied the number of nodes keeping the amount of traffic fixed assuming static routing for simplicity of our work. Finally we observed the percentage of change in wavelength requirements decreases on increasing number of nodes. That is for a large network number of wavelength requirements are large for even number of nodes than odd number of nodes. But this difference is little more for a small network size.

Exact results for a simple epidemic model on a directed network: explorations of a system in a nonequilibrium steady state.

Motivated by fundamental issues in nonequilibrium statistical mechanics, we study the venerable susceptible-infected-susceptible (SIS) model of disease spreading in an idealized, simple setting. Using Monte Carlo and analytic techniques, we consider a fully connected, unidirectional network of odd number of nodes, each having an equal number of in- and out-degrees. With the standard SIS dynamics at high infection rates, this system settles into an active nonequilibrium steady state. We find the exact probability distribution and explore its implications for nonequilibrium statistical mechanics, such as the presence of persistent probability currents.

Asymptotic behavior of orthogonal trigonometric polynomials of semi-integer degree

Orthogonal systems of trigonometric polynomials of semi-integer degree with respect to a weight function w(x) on [0,2π) have been considered firstly by Turetzkii [A.H. Turetzkii, On quadrature formulae that are exact for trigonometric polynomials, East J. Approx. 11 (2005) 337–359 (translation in English from Uchenye Zapiski, Vypusk 1(149), Seria Math. Theory of Functions, Collection of papers, Izdatel’stvo Belgosuniversiteta imeni V.I. Lenina, Minsk, (1959) pp. 31–54)]. Such orthogonal systems are connected with quadrature rules with an even maximal trigonometric degree of exactness (with an odd number of nodes), which have application in numerical integration of 2π-periodic functions. In this paper we study asymptotic behavior of orthogonal trigonometric polynomials of semi-integer degree with respect to a strictly positive weight function satisfying the Lipschitz-Dini condition.

A COMPOUNDING ALGORITHM FOR 1-FAULT TOLERANT BROADCAST NETWORKS

This theoretical paper presents a new construction algorithm for sparse 1-fault tolerant minimal broadcast networks (1-FTMBNs). To construct a large 1-FTMBN with the algorithm, identical nodes in copies of a broadcast network that is not fault tolerant are connected according to the edge structures of two known 1-FTMBNs with a special property. In most cases, computational results with this algorithm produce the best-known bounds on the cardinalities of the edge sets of the sparsest 1-FTMBNs with an odd number of nodes.

Chair-Free Berge Graphs Are Perfect

A graph G is called Berge if neither G nor its complement contains a chordless cycle with an odd number of nodes. The famous Berge's Strong Perfect Graph Conjecture asserts that every Berge graph is perfect. A chair is a graph with nodes {a, b, c, d, e} and edges {ab, bc, cd}, eb. We prove that a Berge graph with no induced chair (chair-free) is perfect or, equivalently, that the Strong Perfect Graph Conjecture is true for chair-free graphs.

Levinson's theorem for Dirac particles.

Levinson's theorem for Dirac particles constrains the sum of the phase shifts at threshold by the total number of bound states of the Dirac equation. Recently, a stronger version of Levinson's theorem has been proven in which the value of the positive- and negative-energy phase shifts are separately constrained by the number of bound states of an appropriate set of Schr\"odinger-like equations. In this work we elaborate on these ideas and show that the stronger form of Levinson's theorem relates the individual phase shifts directly to the number of bound states of the Dirac equation having an even or odd number of nodes. We use a mean-field approximation to Walecka's scalar-vector model to illustrate this stronger form of Levinson's theorem. We show that the assignment of bound states to a particular phase shift should be done, not on the basis of the sign of the bound-state energy, but rather, in terms of the nodal structure (even/odd number of nodes) of the bound state.

Critical graphs, matchings and tours or a hierarchy of relaxations for the travelling salesman problem

A(perfect) 2-matching in a graphG=(V, E) is an assignment of an integer 0, 1 or 2 to each edge of the graph in such a way that the sum over the edges incident with each node is at most (exactly) two. The incidence vector of a Hamiltonian cycle, if one exists inG, is an example of a perfect 2-matching. Fork satisfying 1≦k≦|V|, we letPk denote the problem of finding a perfect 2-matching ofG such that any cycle in the solution contains more thank edges. We call such a matching aperfect Pk-matching. Then fork<l, the problemPk is a relaxation ofP1. Moreover if |V| is odd, thenP1V1–2 is simply the problem of determining whether or notG is Hamiltonian. A graph isPk-critical if it has no perfectPk-matching but whenever any node is deleted the resulting graph does have one. Ifk=|V|, then a graphG=(V, E) isPk-critical if and only if it ishypomatchable (the graph has an odd number of nodes and whatever node is deleted the resulting graph has a perfect matching). We prove the following results: 1. If a graph isPk-critical, then it is alsoPl-critical for all largerl. In particular, for allk, Pk-critical graphs are hypomatchable. 2. A graphG=(V, E) has a perfectPk-matching if and only if for anyX⊆V the number ofPk-critical components inG[V - X] is not greater than |X|. 3. The problemPk can be solved in polynomial time provided we can recognizePk-critical graphs in polynomial time. In addition, we describe a procedure for recognizingPk-critical graphs which is polynomial in the size of the graph and exponential ink.

On Perfect Matchings

Introduction. A graph G = {N, E} is taken to be a finite set of nodes N together with a set of distinct edges E which are unordered pairs of distinct nodes. A matching M of the graph is a subset of the edges E with the property that no two edges of M are incident at a node. A matching M is perfect if every node is incident to an edge of M. For any finite set X, let IXI denote its cardinality. If S c N let G(N S) be the subgraph of G consisting of nodes N S (nodes of N but not S) and all edges of E which only join nodes of N S. For any subgraph H of G, let o(H) be the number of connected components of H each of which consists of an odd number of nodes. If, for some S c N, o(G(N S)) > ISI, we say that the set S is an imperfectable set of nodes of G.