1,073 research outputs found
Conflict-free connection numbers of line graphs
A path in an edge-colored graph is called \emph{conflict-free} if it contains
at least one color used on exactly one of its edges. An edge-colored graph
is \emph{conflict-free connected} if for any two distinct vertices of ,
there is a conflict-free path connecting them. For a connected graph , the
\emph{conflict-free connection number} of , denoted by , is defined
as the minimum number of colors that are required to make conflict-free
connected. In this paper, we investigate the conflict-free connection numbers
of connected claw-free graphs, especially line graphs. We first show that for
an arbitrary connected graph , there exists a positive integer such that
. Secondly, we get the exact value of the conflict-free
connection number of a connected claw-free graph, especially a connected line
graph. Thirdly, we prove that for an arbitrary connected graph and an
arbitrary positive integer , we always have , with only the exception that is isomorphic to a star of order
at least~ and . Finally, we obtain the exact values of ,
and use them as an efficient tool to get the smallest nonnegative integer
such that .Comment: 11 page
Compétences à mobiliser pour la compréhension et l’interprétation de manuels d’histoire du secondaire au Québec
Une analyse de la mise en discours de deux chapitres de deux manuels d’histoire générale conçus pour des élèves québécois de 13-14 ans a été menée afin de connaître quelles compétences ces élèves doivent mobiliser en lecture pour que ces manuels constituent un réel outil d’apprentissage, a fortiori lorsqu’ils se présentent comme des substituts de l’enseignant. Il en ressort que, sans un étayage systématique de la part des enseignants pour assurer la compréhension et l’interprétation des élèves, ces manuels ne peuvent jouer leur rôle, car ils présentent de trop nombreux obstacles pour de jeunes lecteurs.This article presents an analysis of the writing in two chapters in two general history textbooks for Quebec students aged 13-14 years old. The objective was to determine the reading competencies required to use these texts as a real learning tool when presented as a teacher substitute. The results show that, without a systematic presentation by teachers for ensuring student comprehension and interpretation, these textbooks cannot serve their role, since there are too many obstacles for young readers.Se llevó a cabo un análisis del discurso de dos capítulos de dos libros de texto en historia general, concebidos para alumnos quebequenses de 13-14 años para conocer las competencias que estos alumnos deben de movilizar en lectura para que estos libros de texto constituyan una verdadera herramienta de aprendizaje, con más razón cuando pretenden ser sustitutos del docente. Se destaca que, si los docentes no proporcionan sistemáticamente más información a los alumnos para asegurarse de su comprensión y de su interpretación, estos libros de texto no pueden cumplir con las expectativas dado que presentan demasiados obstáculos para los jóvenes lectores
On Metric Dimension of Functigraphs
The \emph{metric dimension} of a graph , denoted by , is the
minimum number of vertices such that each vertex is uniquely determined by its
distances to the chosen vertices. Let and be disjoint copies of a
graph and let be a function. Then a
\emph{functigraph} has the vertex set
and the edge set . We study how
metric dimension behaves in passing from to by first showing that
, if is a connected graph of order
and is any function. We further investigate the metric dimension of
functigraphs on complete graphs and on cycles.Comment: 10 pages, 7 figure
Colourings of cubic graphs inducing isomorphic monochromatic subgraphs
A -bisection of a bridgeless cubic graph is a -colouring of its
vertex set such that the colour classes have the same cardinality and all
connected components in the two subgraphs induced by the colour classes
(monochromatic components in what follows) have order at most . Ban and
Linial conjectured that every bridgeless cubic graph admits a -bisection
except for the Petersen graph. A similar problem for the edge set of cubic
graphs has been studied: Wormald conjectured that every cubic graph with
has a -edge colouring such that the two
monochromatic subgraphs are isomorphic linear forests (i.e. a forest whose
components are paths). Finally, Ando conjectured that every cubic graph admits
a bisection such that the two induced monochromatic subgraphs are isomorphic.
In this paper, we give a detailed insight into the conjectures of Ban-Linial
and Wormald and provide evidence of a strong relation of both of them with
Ando's conjecture. Furthermore, we also give computational and theoretical
evidence in their support. As a result, we pose some open problems stronger
than the above mentioned conjectures. Moreover, we prove Ban-Linial's
conjecture for cubic cycle permutation graphs.
As a by-product of studying -edge colourings of cubic graphs having linear
forests as monochromatic components, we also give a negative answer to a
problem posed by Jackson and Wormald about certain decompositions of cubic
graphs into linear forests.Comment: 33 pages; submitted for publicatio
Speed of synchronization in complex networks of neural oscillators Analytic results based on Random Matrix Theory
We analyze the dynamics of networks of spiking neural oscillators. First, we
present an exact linear stability theory of the synchronous state for networks
of arbitrary connectivity. For general neuron rise functions, stability is
determined by multiple operators, for which standard analysis is not suitable.
We describe a general non-standard solution to the multi-operator problem.
Subsequently, we derive a class of rise functions for which all stability
operators become degenerate and standard eigenvalue analysis becomes a suitable
tool. Interestingly, this class is found to consist of networks of leaky
integrate and fire neurons. For random networks of inhibitory
integrate-and-fire neurons, we then develop an analytical approach, based on
the theory of random matrices, to precisely determine the eigenvalue
distribution. This yields the asymptotic relaxation time for perturbations to
the synchronous state which provides the characteristic time scale on which
neurons can coordinate their activity in such networks. For networks with
finite in-degree, i.e. finite number of presynaptic inputs per neuron, we find
a speed limit to coordinating spiking activity: Even with arbitrarily strong
interaction strengths neurons cannot synchronize faster than at a certain
maximal speed determined by the typical in-degree.Comment: 17 pages, 12 figures, submitted to Chao
Network Extreme Eigenvalue - from Multimodal to Scale-free Network
The extreme eigenvalues of adjacency matrices are important indicators on the
influences of topological structures to collective dynamical behavior of
complex networks. Recent findings on the ensemble averageability of the extreme
eigenvalue further authenticate its sensibility in the study of network
dynamics. Here we determine the ensemble average of the extreme eigenvalue and
characterize the deviation across the ensemble through the discrete form of
random scale-free network. Remarkably, the analytical approximation derived
from the discrete form shows significant improvement over the previous results.
This has also led us to the same conclusion as [Phys. Rev. Lett. 98, 248701
(2007)] that deviation in the reduced extreme eigenvalues vanishes as the
network size grows.Comment: 12 pages, 4 figure
On Graph-Theoretic Identifications of Adinkras, Supersymmetry Representations and Superfields
In this paper we discuss off-shell representations of N-extended
supersymmetry in one dimension, ie, N-extended supersymmetric quantum
mechanics, and following earlier work on the subject codify them in terms of
certain graphs, called Adinkras. This framework provides a method of generating
all Adinkras with the same topology, and so also all the corresponding
irreducible supersymmetric multiplets. We develop some graph theoretic
techniques to understand these diagrams in terms of a relatively small amount
of information, namely, at what heights various vertices of the graph should be
"hung".
We then show how Adinkras that are the graphs of N-dimensional cubes can be
obtained as the Adinkra for superfields satisfying constraints that involve
superderivatives. This dramatically widens the range of supermultiplets that
can be described using the superspace formalism and organizes them. Other
topologies for Adinkras are possible, and we show that it is reasonable that
these are also the result of constraining superfields using superderivatives.
The family of Adinkras with an N-cubical topology, and so also the sequence
of corresponding irreducible supersymmetric multiplets, are arranged in a
cyclical sequence called the main sequence. We produce the N=1 and N=2 main
sequences in detail, and indicate some aspects of the situation for higher N.Comment: LaTeX, 58 pages, 52 illustrations in color; minor typos correcte
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