12 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
Estudi bibliomètric primer trimestre 2014. EETAC
El present document recull les publicacions indexades a la base de dades Scopus durant el període comprès entre el mesos de gener a abril de l’any 2014, escrits per autors pertanyents a l’EETAC. Es presenten les dades recollides segons la font on s’ha publicat, els autors que han publicat, i el tipus de document publicat. S’hi inclou un annex amb la llista de totes les referències bibliogràfiques publicades.El present document recull les publicacions indexades a la base de dades Scopus durant el període comprès entre el mesos de gener a abril de l’any 2014, escrits per autors pertanyents a l’EETAC. Es presenten les dades recollides segons la font on s’ha publicat, els autors que han publicat, i el tipus de document publicat. S’hi inclou un annex amb la llista de totes les referències bibliogràfiques publicades.Postprint (published version
Estudi bibliomètric any 2014. Campus del Baix Llobregat: EETAC i ESAB
En el present informe s’analitza la producció científica de les dues escoles del Campus del Baix Llobregat, l’Escola d’Enginyeria de Telecomunicació i Aerospacial de Castelldefels (EETAC) i l’Escola Superior d’Agricultura de Barcelona (ESAB) durant el 2014.Postprint (author’s final draft
On symmetric association schemes and associated quotient-polynomial graphs
Let denote an undirected, connected, regular graph with vertex set
, adjacency matrix , and distinct eigenvalues. Let denote the subalgebra of Mat
generated by . We refer to as the {\it adjacency algebra} of
. In this paper we investigate algebraic and combinatorial structure of
for which the adjacency algebra is closed under
Hadamard multiplication. In particular, under this simple assumption, we show
the following: (i) has a standard basis ;
(ii) for every vertex there exists identical distance-faithful intersection
diagram of with cells; (iii) the graph is
quotient-polynomial; and (iv) if we pick then
has distinct eigenvalues if and only if
spanspan. We describe the
combinatorial structure of quotient-polynomial graphs with diameter and
distinct eigenvalues. As a consequence of the technique from the paper we give
an algorithm which computes the number of distinct eigenvalues of any Hermitian
matrix using only elementary operations. When such a matrix is the adjacency
matrix of a graph , a simple variation of the algorithm allow us to
decide wheter is distance-regular or not. In this context, we also
propose an algorithm to find which distance- matrices are polynomial in ,
giving also these polynomials.Comment: 22 pages plus 4 pages of reference
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
Algebraic Characterizations of Regularity Properties in Bipartite Graphs
Regular and distance-regular characterizations of general graphs are well-known. In particular, the spectral excess theorem states that a connected graph GG is distance-regular if and only if its spectral excess (a number that can be computed from the spectrum) equals the average excess (the mean of the numbers of vertices at extremal distance from every vertex). The aim of this paper is to derive new characterizations of regularity and distance-regularity for the more restricted family of bipartite graphs. In this case, some characterizations of (bi)regular bipartite graphs are given in terms of the mean degrees in every partite set and the Hoffman polynomial. Moreover, it is shown that the conditions for having distance-regularity in such graphs can be relaxed when compared with general graphs. Finally, a new version of the spectral excess theorem for bipartite graphs is presented
Algebraic Characterizations of Regularity Properties in Bipartite Graphs
Regular and distance-regular characterizations of general graphs are well-known. In particular, the spectral excess theorem states that a connected graph GG is distance-regular if and only if its spectral excess (a number that can be computed from the spectrum) equals the average excess (the mean of the numbers of vertices at extremal distance from every vertex). The aim of this paper is to derive new characterizations of regularity and distance-regularity for the more restricted family of bipartite graphs. In this case, some characterizations of (bi)regular bipartite graphs are given in terms of the mean degrees in every partite set and the Hoffman polynomial. Moreover, it is shown that the conditions for having distance-regularity in such graphs can be relaxed when compared with general graphs. Finally, a new version of the spectral excess theorem for bipartite graphs is presented
Corrigendum to "Algebraic characterizations of regularity properties in bipartite graphs" Eur. J. Combin. 34 (2013) 1223-1231
Corrigendum d'un article anteriorment publicat. DOI: 10.1016/j.ejc.2013.05.00