9,484 research outputs found
A characterization and an application of weight-regular partitions of graphs
A natural generalization of a regular (or equitable) partition of a graph,
which makes sense also for non-regular graphs, is the so-called weight-regular
partition, which gives to each vertex a weight that equals the
corresponding entry of the Perron eigenvector . This
paper contains three main results related to weight-regular partitions of a
graph. The first is a characterization of weight-regular partitions in terms of
double stochastic matrices. Inspired by a characterization of regular graphs by
Hoffman, we also provide a new characterization of weight-regularity by using a
Hoffman-like polynomial. As a corollary, we obtain Hoffman's result for regular
graphs. In addition, we show an application of weight-regular partitions to
study graphs that attain equality in the classical Hoffman's lower bound for
the chromatic number of a graph, and we show that weight-regularity provides a
condition under which Hoffman's bound can be improved
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
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
Perfect domination in regular grid graphs
We show there is an uncountable number of parallel total perfect codes in the
integer lattice graph of . In contrast, there is just one
1-perfect code in and one total perfect code in
restricting to total perfect codes of rectangular grid graphs (yielding an
asymmetric, Penrose, tiling of the plane). We characterize all cycle products
with parallel total perfect codes, and the -perfect and
total perfect code partitions of and , the former
having as quotient graph the undirected Cayley graphs of with
generator set . For , generalization for 1-perfect codes is
provided in the integer lattice of and in the products of cycles,
with partition quotient graph taken as the undirected Cayley graph
of with generator set .Comment: 16 pages; 11 figures; accepted for publication in Austral. J. Combi
Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing
We consider sequences of graphs and define various notions of convergence
related to these sequences: ``left convergence'' defined in terms of the
densities of homomorphisms from small graphs into the graphs of the sequence,
and ``right convergence'' defined in terms of the densities of homomorphisms
from the graphs of the sequence into small graphs; and convergence in a
suitably defined metric.
In Part I of this series, we show that left convergence is equivalent to
convergence in metric, both for simple graphs, and for graphs with nodeweights
and edgeweights. One of the main steps here is the introduction of a
cut-distance comparing graphs, not necessarily of the same size. We also show
how these notions of convergence provide natural formulations of Szemeredi
partitions, sampling and testing of large graphs.Comment: 57 pages. See also http://research.microsoft.com/~borgs/. This
version differs from an earlier version from May 2006 in the organization of
the sections, but is otherwise almost identica
Quasi-randomness and algorithmic regularity for graphs with general degree distributions
We deal with two intimately related subjects: quasi-randomness and regular partitions. The purpose of the concept of quasi-randomness is to express how much a given graph âresemblesâ a random one. Moreover, a regular partition approximates a given graph by a bounded number of quasi-random graphs. Regarding quasi-randomness, we present a new spectral characterization of low discrepancy, which extends to sparse graphs. Concerning regular partitions, we introduce a concept of regularity that takes into account vertex weights, and show that if satisfies a certain boundedness condition, then admits a regular partition. In addition, building on the work of Alon and Naor [Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), Chicago, IL, ACM, New York, 2004, pp. 72â80], we provide an algorithm that computes a regular partition of a given (possibly sparse) graph in polynomial time. As an application, we present a polynomial time approximation scheme for MAX CUT on (sparse) graphs without âdense spots.
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