3,622 research outputs found
Guides and Shortcuts in Graphs
The geodesic structure of a graphs appears to be a very rich structure. There are many ways to describe this structure, each of which captures only some aspects. Ternary algebras are for this purpose very useful and have a long tradition. We study two instances: signpost systems, and a special case of which, step systems. Signpost systems were already used to characterize graph classes. Here we use these for the study of the geodesic structure of a spanning subgraph F with respect to its host graph G. Such a signpost system is called a guide to (F,G). Our main results are: the characterization of the step system of a cycle, the characterization of guides for spanning trees and hamiltonian cycles
Ising model in small-world networks
The Ising model in small-world networks generated from two- and
three-dimensional regular lattices has been studied. Monte Carlo simulations
were carried out to characterize the ferromagnetic transition appearing in
these systems. In the thermodynamic limit, the phase transition has a
mean-field character for any finite value of the rewiring probability p, which
measures the disorder strength of a given network. For small values of p, both
the transition temperature and critical energy change with p as a power law. In
the limit p -> 0, the heat capacity at the transition temperature diverges
logarithmically in two-dimensional (2D) networks and as a power law in 3D.Comment: 6 pages, 7 figure
Slow relaxation in the Ising model on a small-world network with strong long-range interactions
We consider the Ising model on a small-world network, where the long-range
interaction strength is in general different from the local interaction
strength , and examine its relaxation behaviors as well as phase
transitions. As is raised from zero, the critical temperature also
increases, manifesting contributions of long-range interactions to ordering.
However, it becomes saturated eventually at large values of and the
system is found to display very slow relaxation, revealing that ordering
dynamics is inhibited rather than facilitated by strong long-range
interactions. To circumvent this problem, we propose a modified updating
algorithm in Monte Carlo simulations, assisting the system to reach equilibrium
quickly.Comment: 5 pages, 5 figure
Self-avoiding walks and connective constants in small-world networks
Long-distance characteristics of small-world networks have been studied by
means of self-avoiding walks (SAW's). We consider networks generated by
rewiring links in one- and two-dimensional regular lattices. The number of
SAW's was obtained from numerical simulations as a function of the number
of steps on the considered networks. The so-called connective constant,
, which characterizes the long-distance
behavior of the walks, increases continuously with disorder strength (or
rewiring probability, ). For small , one has a linear relation , and being constants dependent on the underlying
lattice. Close to one finds the behavior expected for random graphs. An
analytical approach is given to account for the results derived from numerical
simulations. Both methods yield results agreeing with each other for small ,
and differ for close to 1, because of the different connectivity
distributions resulting in both cases.Comment: 7 pages, 5 figure
Abstracts of theses and related literature indicating current trends in arithmetic for the academically talented elementary school child written between 1957 and 1961
Thesis (Ed.M.)--Boston Universit
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