General information

From** June 29 to July 3, 2015** at the CRM (location)

Timetable

List of participants

Grants available

Download the poster here

Goals

Combinatorial
matrix theory is a rich branch of matrix theory concerned with the interplay of
combinatorics/graph theory and matrix theory/linear algebra. It's a two way
street, with linear algebra providing a means to prove combinatorial theorems,
and combinatorics providing more detailed and refined information in linear
algebra. Moreover, combinatorial properties of matrices are studied based on
qualitative rather than quantitative information, so that the ideas developed
can provide consistent information about a model even when the data is
incomplete or inaccurate. The theory behind qualitative methods can also
contribute to the development of effective quantitative matrix methods.

Particular
topics lectured in this summer school will include sign pattern
matrices, minimum rank and its distribution, boundary value problems on finite
networks, the group inverse for the Laplacian matrix of a graph and bounds on
the spectral radius of the Laplacian matrix.

Scientific Committee

Andrés M. Encinas, Universitat Politècnica de Catalunya

Carlos Da Fonseca, Kuwait University

Margarida Mitjana, Universitat Politècnica de Catalunya

Each lecturer will give 5 hours
course:

*Richard A. Brualdi*,
University of Wisconsin-Madison

*Angeles Carmona Mejías*,
Universitat Politècnica de Catalunya-BarcelonaTech

*Stephen J. Kirkland*, University of Manitoba

*Dragan Stevanovic*, Serbian Academy of Sciences and Arts (SANU)

*Pauline van den Driessche*, University of Victoria

**LECTURE
ABSTRACTS**

*Richard A. Brualdi*

*Combinatorial Matrix Theory*

Combinatorial
Matrix Theory (CMT) is concerned with the interplay of combinatorics/graph
theory and matrix theory/linear algebra. It's a two way street, with linear
algebra providing a means to prove combinatorial theorems, and combinatorics
providing more detailed and refined information in linear algebra. In addition,
in CMT classes of matrices (adjacency matrices of bipartite graphs, or in the
symmetric case, graphs) are investigated with the goal of understanding such
classes and investigation of combinatorial or linear algebraic invariants over
the class. In these lectures I plan to discuss many aspects of CMT.

*Angeles Carmona*

*Boundary
value problems on Finite Networks*

The aim of these
lectures is to analyze self–adjoint boundary value problems on finite networks.
We start with the description of the basic difference operators: the
derivative, gradient, divergence, curl and laplacian, or more generally,
Schrodinger operators. Moreover, we prove that the above operators satisfy
analogue properties to those exhibited by their continuous counterpart. The
next step is to define the discrete analogue of a manifold with boundary, which
includes the concept of outer normal field. Then, we prove the Green Identities
in order to establish the variational formulation of boundary value problems.
Then, we focus on some aspects of discrete Potential Theory by proving the discrete
version of the Dirichlet, the maximum and the condenser principles. In this
framework, another useful tool is the concept of Resolvent Kernel associated
with a boundary value problem. So, we define the discrete analogous of the
Green and Poisson Kernels and we establish its main properties and relations.
Then, we also deal with the Dirichlet to Neumann map and study its relation
with the Poisson and Green kernels. Finally, we consider some applications to
Matrix Theory and to Organic Chemistry, as the M–inverse problem and the
Kirchhoff Index computation, respectively.

*Stephen Kirkland*

*The Group
Inverse for the Laplacian Matrix of a Graph*

Laplacian
matrices for undirected graphs have received a good deal of attention, in part
because the spectral properties of the Laplacian matrix are related to a number
of features of interest of the underlying graph. It turns out that a certain
generalised inverse - the group inverse - of a Laplacian matrix also carries
information about the graph in question. This series of lectures will explore
the group inverse of the Laplacian matrix and its relationship with graph
structure. Connections with algebraic connectivity and resistance distance will
be made, and the computation of the group inverse of a Laplacian matrix will
also be considered from a numerical viewpoint.

*Dragan
Stevanovic*

*Spectral
radius of graphs*

Eigenvalues and
eigenvectors of graph matrices have become standard mathematical tools nowadays
due to their wide applicability in network analysis and computer
science, with the most prominent graph matrices being the adjacency and
the Laplacian matrix. In these lectures I will survey lower and upper bounds on
the spectral radius of adjacency and Laplacian matrices, with focus on the
proof techniques and on common properties of the forms of the bounds. Some
interesting approximate formulas for the spectral radius of adjacency matrix
will be discussed as well.

*Pauline van den Driessche*

*Sign Pattern Matrices*

An *n* x *n* sign
pattern (matrix) *A** = [*α*_{ij}] has entries
from {+, -, 0} with an associated sign pattern class of real *n* x *n* matrices
{*A* = [*a*_{ij}] : sign(*a*_{ij} ) = *α*_{ij} for
all *i, j*}. These lectures will survey some important classes of
sign patterns, including sign patterns that allow all possible spectra, those
that allow all possible inertias, those that allow stability, and those that
may give rise to Hopf bifurcation in associated dynamical systems. Then some
classes will be explored in more detail using techniques from matrix theory,
graph theory and analysis, and open problems will be suggested.* *

Registration

** ****REGISTRATION FEE**

**Junior: 150€**

**Senior: 300€**

Registration includes: Documentation package, lunch (during the days of the advanced course, from Monday to Friday), social dinner, and coffee breaks.

Deadline for registration: **June 21, 2015**

The number of participants** **is limited. See the grants available below.

Please, access the on-line registration process by clicking one of the REGISTER buttons

Grants

A limited number of grants (lodging and/or reduced registration) are available for PhD students and recent PhD holders (who received their PhDs no more than two years ago).

Check the details here

Lodging

For lodging in the area please click

here

For off-campus and family accommodation click

here

Acknowledgments

Partial support for students attending this advanced
course has been received from Elsevier, publisher of the journal "LinearAlgebra and its Applications".

This advanced course has also received support from the Societat Catalana de Matemàtiques

Contact

For further information contact the programme coordinator Neus Portet (nportet@crm.cat)