77 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
An application of Hoffman graphs for spectral characterizations of graphs
In this paper, we present the first application of Hoffman graphs for
spectral characterizations of graphs. In particular, we show that the
-clique extension of the -grid is determined by its
spectrum when is large enough. This result will help to show that the
Grassmann graph is determined by its intersection numbers as a
distance regular graph, if is large enough
Enumeration of cospectral and coinvariant graphs
We present enumeration results on the number of connected graphs up to 10
vertices for which there is at least one other graph with the same spectrum (a
cospectral mate), or at least one other graph with the same Smith normal form
(coinvariant mate) with respect to several matrices associated to a graph. The
present data give some indication that possibly the Smith normal form of the
distance Laplacian and the signless distance Laplacian matrices could be a
finer invariant to distinguish graphs in cases where other algebraic
invariants, such as those derived from the spectrum, fail. Finally, we show a
new graph characterization using the Smith normal form of the signless distance
Laplacian matrix
Some spectral and quasi-spectral characterizations of distance-regular graphs
© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In this paper we consider the concept of preintersection numbers of a graph. These numbers are determined by the spectrum of the adjacency matrix of the graph, and generalize the intersection numbers of a distance-regular graph. By using the preintersection numbers we give some new spectral and quasi-spectral characterizations of distance-regularity, in particular for graphs with large girth or large odd-girth. (C) 2016 Published by Elsevier Inc.Peer ReviewedPostprint (author's final draft
Spectral Bounds for the Connectivity of Regular Graphs with Given Order
The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a
graph are measures of its connectivity. These eigenvalues can be used to
analyze the robustness, resilience, and synchronizability of networks, and are
related to connectivity attributes such as the vertex- and edge-connectivity,
isoperimetric number, and characteristic path length. In this paper, we present
two upper bounds for the second-largest eigenvalues of regular graphs and
multigraphs of a given order which guarantee a desired vertex- or
edge-connectivity. The given bounds are in terms of the order and degree of the
graphs, and hold with equality for infinite families of graphs. These results
answer a question of Mohar.Comment: 24 page
Descriptive complexity of controllable graphs
Let be a graph on vertices with adjacency matrix , and let
be the all-ones vector. We call controllable if the set of
vectors spans the whole
space . We characterize the isomorphism problem of controllable
graphs in terms of other combinatorial, geometric and logical problems. We also
describe a polynomial time algorithm for graph isomorphism that works for
almost all graphs.Comment: 14 page
On the diameter and zero forcing number of some graph classes in the Johnson, Grassmann and Hamming association scheme
We establish the diameter of generalized Grassmann graphs and the zero
forcing number of some generalized Johnson graphs, generalized Grassmann graphs
and the Hamming graphs. Our work extends several previously known results
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
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