15,094 research outputs found
The limit points of the top and bottom eigenvalues of regular graphs
We prove that for each the set of all limit points of the second
largest eigenvalue of growing sequences of -regular graphs is
. A similar argument shows that the set of all limit points of
the smallest eigenvalue of growing sequences of -regular graphs with growing
(odd) girth is . The more general question of identifying
all vectors which are limit points of the vectors of the top eigenvalues of
sequences of -regular graphs is considered as well. As a by product, in the
study of discrete counterpart of the "scarring" phenomenon observed in the
investigation of quantum ergodicity on manifolds, our technique provides a
method to construct -regular almost Ramanujan graphs with large girth and
localized eigenvectors corresponding to eigenvalues larger than ,
strengthening a result of Alon, Ganguly, and Srivastava
Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics
Materials science and the study of the electronic properties of solids are a
major field of interest in both physics and engineering. The starting point for
all such calculations is single-electron, or non-interacting, band structure
calculations, and in the limit of strong on-site confinement this can be
reduced to graph-like tight-binding models. In this context, both
mathematicians and physicists have developed largely independent methods for
solving these models. In this paper we will combine and present results from
both fields. In particular, we will discuss a class of lattices which can be
realized as line graphs of other lattices, both in Euclidean and hyperbolic
space. These lattices display highly unusual features including flat bands and
localized eigenstates of compact support. We will use the methods of both
fields to show how these properties arise and systems for classifying the
phenomenology of these lattices, as well as criteria for maximizing the gaps.
Furthermore, we will present a particular hardware implementation using
superconducting coplanar waveguide resonators that can realize a wide variety
of these lattices in both non-interacting and interacting form
Spectra of "Real-World" Graphs: Beyond the Semi-Circle Law
Many natural and social systems develop complex networks, that are usually
modelled as random graphs. The eigenvalue spectrum of these graphs provides
information about their structural properties. While the semi-circle law is
known to describe the spectral density of uncorrelated random graphs, much less
is known about the eigenvalues of real-world graphs, describing such complex
systems as the Internet, metabolic pathways, networks of power stations,
scientific collaborations or movie actors, which are inherently correlated and
usually very sparse. An important limitation in addressing the spectra of these
systems is that the numerical determination of the spectra for systems with
more than a few thousand nodes is prohibitively time and memory consuming.
Making use of recent advances in algorithms for spectral characterization, here
we develop new methods to determine the eigenvalues of networks comparable in
size to real systems, obtaining several surprising results on the spectra of
adjacency matrices corresponding to models of real-world graphs. We find that
when the number of links grows as the number of nodes, the spectral density of
uncorrelated random graphs does not converge to the semi-circle law.
Furthermore, the spectral densities of real-world graphs have specific features
depending on the details of the corresponding models. In particular, scale-free
graphs develop a triangle-like spectral density with a power law tail, while
small-world graphs have a complex spectral density function consisting of
several sharp peaks. These and further results indicate that the spectra of
correlated graphs represent a practical tool for graph classification and can
provide useful insight into the relevant structural properties of real
networks.Comment: 14 pages, 9 figures (corrected typos, added references) accepted for
Phys. Rev.
Eigenvalue interlacing and weight parameters of graphs
Eigenvalue interlacing is a versatile technique for deriving results in
algebraic combinatorics. In particular, it has been successfully used for
proving a number of results about the relation between the (adjacency matrix or
Laplacian) spectrum of a graph and some of its properties. For instance, some
characterizations of regular partitions, and bounds for some parameters, such
as the independence and chromatic numbers, the diameter, the bandwidth, etc.,
have been obtained. For each parameter of a graph involving the cardinality of
some vertex sets, we can define its corresponding weight parameter by giving
some "weights" (that is, the entries of the positive eigenvector) to the
vertices and replacing cardinalities by square norms. The key point is that
such weights "regularize" the graph, and hence allow us to define a kind of
regular partition, called "pseudo-regular," intended for general graphs. Here
we show how to use interlacing for proving results about some weight parameters
and pseudo-regular partitions of a graph. For instance, generalizing a
well-known result of Lov\'asz, it is shown that the weight Shannon capacity
of a connected graph \G, with vertices and (adjacency matrix)
eigenvalues , satisfies \Theta\le
\Theta^* \le \frac{\|\vecnu\|^2}{1-\frac{\lambda_1}{\lambda_n}} where
is the (standard) Shannon capacity and \vecnu is the positive
eigenvector normalized to have smallest entry 1. In the special case of regular
graphs, the results obtained have some interesting corollaries, such as an
upper bound for some of the multiplicities of the eigenvalues of a
distance-regular graph. Finally, some results involving the Laplacian spectrum
are derived. spectrum are derived
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
Circular Coloring of Random Graphs: Statistical Physics Investigation
Circular coloring is a constraints satisfaction problem where colors are
assigned to nodes in a graph in such a way that every pair of connected nodes
has two consecutive colors (the first color being consecutive to the last). We
study circular coloring of random graphs using the cavity method. We identify
two very interesting properties of this problem. For sufficiently many color
and sufficiently low temperature there is a spontaneous breaking of the
circular symmetry between colors and a phase transition forwards a
ferromagnet-like phase. Our second main result concerns 5-circular coloring of
random 3-regular graphs. While this case is found colorable, we conclude that
the description via one-step replica symmetry breaking is not sufficient. We
observe that simulated annealing is very efficient to find proper colorings for
this case. The 5-circular coloring of 3-regular random graphs thus provides a
first known example of a problem where the ground state energy is known to be
exactly zero yet the space of solutions probably requires a full-step replica
symmetry breaking treatment.Comment: 19 pages, 8 figures, 3 table
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