3,345 research outputs found
Some applications of linear algebra in spectral graph theory
The application of the theory of matrices and eigenvalues to combinatorics is cer-
tainly not new. In the present work the starting point is a theorem that concerns the
eigenvalues of partitioned matrices. Interlacing yields information on subgraphs of
a graph, and the way such subgraphs are embedded. In particular, one gets bounds
on extremal substructures. Applications of this theorem and of some known matrix
theorems to matrices associated to graphs lead to new results. For instance, some
characterizations of regular partitions, and bounds for some parameters, such as
the independence and chromatic numbers, the diameter, the bandwidth, etc. This
master thesis is a contribution to the area of algebraic graph theory and the study
of some generalizations of regularity in bipartite graphs.
In Chapter 1 we recall some basic concepts and results from graph theory and linear
algebra.
Chapter 2 presents some simple but relevant results on graph spectra concerning
eigenvalue interlacing. Most of the previous results that we use were obtained by
Haemers in [33]. In that work, the author gives bounds for the size of a maximal
(co)clique, the chromatic number, the diameter and the bandwidth in terms of the
eigenvalues of the standard adjacency matrix or the Laplacian matrix. He also nds
some inequalities and regularity results concerning the structure of graphs.
The work initiated by Fiol [26] in this area leads us to Chapter 3. The discussion
goes along the same spirit, but in this case eigenvalue interlacing is used for proving
results about some weight parameters and weight-regular partitions of a graph. In
this master thesis a new observation leads to a greatly simpli ed notation of the
results related with weight-partitions. We nd an upper bound for the weight
independence number in terms of the minimum degree.
Special attention is given to regular bipartite graphs, in fact, in Chapter 4 we
contribute with an algebraic characterization of regularity properties in bipartite
graphs. Our rst approach to regularity in bipartite graphs comes from the study of
its spectrum. We characterize these graphs using eigenvalue interlacing and we pro-
vide an improved bound for biregular graphs inspired in Guo's inequality. We prove
a condition for existence of a k-dominating set in terms of its Laplacian eigenvalues.
In particular, we give an upper bound on the sum of the rst Laplacian eigenvalues
of a k-dominating set and generalize a Guo's result for these structures. In terms
of predistance polynomials, we give a result that can be seen as the biregular coun-
terpart of Ho man's Theorem. Finally, we also provide new characterizations of
bipartite graphs inspired in the notion of distance-regularity.
In Chapter 5 we describe some ideas to work with a result from linear algebra known
as the Rayleigh's principle. We observe that the clue is to make the \right choice"
of the eigenvector that is used in Rayleigh's principle. We can use this method
1
to give a spectral characterization of regular and biregular partitions. Applying
this technique, we also derive an alternative proof for the upper bound of the
independence number obtained by Ho man (Chapter 2, Theorem 1.2).
Finally, in Chapter 6 other related new results and some open problems are pre-
sented
On the Spectral Gap of a Quantum Graph
We consider the problem of finding universal bounds of "isoperimetric" or
"isodiametric" type on the spectral gap of the Laplacian on a metric graph with
natural boundary conditions at the vertices, in terms of various analytical and
combinatorial properties of the graph: its total length, diameter, number of
vertices and number of edges. We investigate which combinations of parameters
are necessary to obtain non-trivial upper and lower bounds and obtain a number
of sharp estimates in terms of these parameters. We also show that, in contrast
to the Laplacian matrix on a combinatorial graph, no bound depending only on
the diameter is possible. As a special case of our results on metric graphs, we
deduce estimates for the normalised Laplacian matrix on combinatorial graphs
which, surprisingly, are sometimes sharper than the ones obtained by purely
combinatorial methods in the graph theoretical literature
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
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
On the spectrum of hypergraphs
Here we study the spectral properties of an underlying weighted graph of a
non-uniform hypergraph by introducing different connectivity matrices, such as
adjacency, Laplacian and normalized Laplacian matrices. We show that different
structural properties of a hypergrpah, can be well studied using spectral
properties of these matrices. Connectivity of a hypergraph is also investigated
by the eigenvalues of these operators. Spectral radii of the same are bounded
by the degrees of a hypergraph. The diameter of a hypergraph is also bounded by
the eigenvalues of its connectivity matrices. We characterize different
properties of a regular hypergraph characterized by the spectrum. Strong
(vertex) chromatic number of a hypergraph is bounded by the eigenvalues.
Cheeger constant on a hypergraph is defined and we show that it can be bounded
by the smallest nontrivial eigenvalues of Laplacian matrix and normalized
Laplacian matrix, respectively, of a connected hypergraph. We also show an
approach to study random walk on a (non-uniform) hypergraph that can be
performed by analyzing the spectrum of transition probability operator which is
defined on that hypergraph. Ricci curvature on hypergraphs is introduced in two
different ways. We show that if the Laplace operator, , on a hypergraph
satisfies a curvature-dimension type inequality
with and then any non-zero eigenvalue of can be bounded below by . Eigenvalues of a normalized Laplacian operator defined on a connected
hypergraph can be bounded by the Ollivier's Ricci curvature of the hypergraph
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