1,428 research outputs found
Spectral radius of finite and infinite planar graphs and of graphs of bounded genus
It is well known that the spectral radius of a tree whose maximum degree is
cannot exceed . In this paper we derive similar bounds for
arbitrary planar graphs and for graphs of bounded genus. It is proved that a
the spectral radius of a planar graph of maximum vertex degree
satisfies . This result is
best possible up to the additive constant--we construct an (infinite) planar
graph of maximum degree , whose spectral radius is . This
generalizes and improves several previous results and solves an open problem
proposed by Tom Hayes. Similar bounds are derived for graphs of bounded genus.
For every , these bounds can be improved by excluding as a
subgraph. In particular, the upper bound is strengthened for 5-connected
graphs. All our results hold for finite as well as for infinite graphs.
At the end we enhance the graph decomposition method introduced in the first
part of the paper and apply it to tessellations of the hyperbolic plane. We
derive bounds on the spectral radius that are close to the true value, and even
in the simplest case of regular tessellations of type we derive an
essential improvement over known results, obtaining exact estimates in the
first order term and non-trivial estimates for the second order asymptotics
Spectral radii of sparse random matrices
We establish bounds on the spectral radii for a large class of sparse random
matrices, which includes the adjacency matrices of inhomogeneous
Erd\H{o}s-R\'enyi graphs. Our error bounds are sharp for a large class of
sparse random matrices. In particular, for the Erd\H{o}s-R\'enyi graph
, our results imply that the smallest and second-largest eigenvalues
of the adjacency matrix converge to the edges of the support of the asymptotic
eigenvalue distribution provided that . Together with the
companion paper [3], where we analyse the extreme eigenvalues in the
complementary regime , this establishes a crossover in the
behaviour of the extreme eigenvalues around . Our results also
apply to non-Hermitian sparse random matrices, corresponding to adjacency
matrices of directed graphs. The proof combines (i) a new inequality between
the spectral radius of a matrix and the spectral radius of its nonbacktracking
version together with (ii) a new application of the method of moments for
nonbacktracking matrices
Walks and the spectral radius of graphs
We give upper and lower bounds on the spectral radius of a graph in terms of
the number of walks. We generalize a number of known results.Comment: Corrections were made in Theorems 5 and 11 (the new numbers are
different), following a remark of professor Yaoping Ho
A transfer principle and applications to eigenvalue estimates for graphs
In this paper, we prove a variant of the Burger-Brooks transfer principle
which, combined with recent eigenvalue bounds for surfaces, allows to obtain
upper bounds on the eigenvalues of graphs as a function of their genus. More
precisely, we show the existence of a universal constants such that the
-th eigenvalue of the normalized Laplacian of a graph
of (geometric) genus on vertices satisfies where denotes the maximum valence of
vertices of the graph. This result is tight up to a change in the value of the
constant , and improves recent results of Kelner, Lee, Price and Teng on
bounded genus graphs.
To show that the transfer theorem might be of independent interest, we relate
eigenvalues of the Laplacian on a metric graph to the eigenvalues of its simple
graph models, and discuss an application to the mesh partitioning problem,
extending pioneering results of Miller-Teng-Thurston-Vavasis and Spielman-Tang
to arbitrary meshes.Comment: Major revision, 16 page
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