3,685 research outputs found
Properties of dense partially random graphs
We study the properties of random graphs where for each vertex a {\it
neighbourhood} has been previously defined. The probability of an edge joining
two vertices depends on whether the vertices are neighbours or not, as happens
in Small World Graphs (SWGs). But we consider the case where the average degree
of each node is of order of the size of the graph (unlike SWGs, which are
sparse). This allows us to calculate the mean distance and clustering, that are
qualitatively similar (although not in such a dramatic scale range) to the case
of SWGs. We also obtain analytically the distribution of eigenvalues of the
corresponding adjacency matrices. This distribution is discrete for large
eigenvalues and continuous for small eigenvalues. The continuous part of the
distribution follows a semicircle law, whose width is proportional to the
"disorder" of the graph, whereas the discrete part is simply a rescaling of the
spectrum of the substrate. We apply our results to the calculation of the
mixing rate and the synchronizability threshold.Comment: 14 pages. To be published in Physical Review
A theory of spectral partitions of metric graphs
We introduce an abstract framework for the study of clustering in metric
graphs: after suitably metrising the space of graph partitions, we restrict
Laplacians to the clusters thus arising and use their spectral gaps to define
several notions of partition energies; this is the graph counterpart of the
well-known theory of spectral minimal partitions on planar domains and includes
the setting in [Band \textit{et al}, Comm.\ Math.\ Phys.\ \textbf{311} (2012),
815--838] as a special case. We focus on the existence of optimisers for a
large class of functionals defined on such partitions, but also study their
qualitative properties, including stability, regularity, and parameter
dependence. We also discuss in detail their interplay with the theory of nodal
partitions. Unlike in the case of domains, the one-dimensional setting of
metric graphs allows for explicit computation and analytic -- rather than
numerical -- results. Not only do we recover the main assertions in the theory
of spectral minimal partitions on domains, as studied in [Conti \textit{et al},
Calc.\ Var.\ \textbf{22} (2005), 45--72; Helffer \textit{et al}, Ann.\ Inst.\
Henri Poincar\'e Anal.\ Non Lin\'eaire \textbf{26} (2009), 101--138], but we
can also generalise some of them and answer (the graph counterparts of) a few
open questions
Nodal domains, spectral minimal partitions, and their relation to Aharonov-Bohm operators
This survey is a short version of a chapter written by the first two authors
in the book [A. Henrot, editor. Shape optimization and spectral theory. Berlin:
De Gruyter, 2017] (where more details and references are given) but we have
decided here to put more emphasis on the role of the Aharonov-Bohm operators
which appear to be a useful tool coming from physics for understanding a
problem motivated either by spectral geometry or dynamics of population.
Similar questions appear also in Bose-Einstein theory. Finally some open
problems which might be of interest are mentioned.Comment: arXiv admin note: substantial text overlap with arXiv:1506.0724
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