A.M. Encinas

Estimation of Fekete points

We aim here at presenting a new procedure to numerically estimate the Fekete points of a wide variety of compact sets in R^3. We understand the Fekete point problem in terms of the identification of near equilibrium configurations for a potential energy that depends on the relative position of N particles. The compact sets for which our procedure works are basically the finite union of piecewise regular surfaces and curves. In order to determine a good initial configuration to start the search of the Fekete points of these objects, we construct a sequence of approximating regular surfaces. Our algorithm is based on the concept of disequilibrium degree, which is defined from a physical interpretation of the behavior of a system of particles when they search for a minimum energy configuration. Moreover, the algorithm is efficient and robust independently of the considered compact set as well as of the kernel used to define the energy. The numerical experimentation carried out suggests that a local minimum can be localized with a computational cost of order less than N^3.

Potential Theory for Schrödinger operators on finite networks

We aim here at analyzing the fundamental properties of positive semidefinite Schrodinger operators on networks. We show that such operators correspond to perturbations of the combinatorial Laplacian through 0-order terms that can be totally negative on a proper subset of the network. In addition, we prove that these discrete operators have analogous properties to the ones of elliptic second order operators on Riemannian manifolds, namely the monotonicity, the minimum principle, the variational treatment of Dirichlet problems and the condenser principle. Unlike the continuous case, a discrete Schrodinger operator can be interpreted as an integral operator and therefore a discrete Potential Theory with respect to its associated kernel can be built. We prove that the Schrodinger kernel satisfies enough principles to assure the existence of equilibrium measures for any proper subset. These measures are used to obtain systematic expressions of the Green and Poisson kernels associated with Dirichlet problems.

Computational cost of the Fekete problem I: The Forces Method on the 2-sphere

Here, we study the computational complexity of the Fekete point problem. Namely, we give an exhaustive description of the main properties of an algorithm for the minimization of the logarithmic potential energy on the 2-sphere, and we characterize the probability distribution of the cost of the different minima. In particular, we show that a local minimum can be found with an average cost of about O(N^2^.^8).

Shortest Paths in Distance-regular Graphs

We aim here to introduce a new point of view of the Laplacian of a graph, ?. With this purpose in mind, we consider L as a kernel on the finite space V(?), in the context of the Potential Theory. Then we prove that L is a nice kernel, since it verifies some fundamental properties such as maximum and energy principles and the equilibrium principle on any proper subset of V(?). If ? is a proper set of a suitable host graph, then the equilibrium problem for ? can be solved and the number of the different components of its equilibrium measure leads to a bound on the diameter of ?. In particular, we obtain the structure of the shortest paths of a distance-regular graph. As a consequence, we find the intersection array in terms of the equilibrium measure. Finally, we give a new characterization of strongly regular graphs.

Solving Dirichlet and Poisson problems on graphs by means of equilibrium measures

We aim here at obtaining an explicit expression of the solution of the Dirichlet and Poisson problems on graphs. To this end, we consider the Laplacian of a graph as a kernel on the vertex set, V , in the framework of Potential Theory. Then, the properties of such a kernel allow us to obtain for each proper vertex subset the equilibrium measure that solves the so-called equilibrium problem . As a consequence, the Green function of the Dirichlet problems, the generalized Green function of the Poisson problems and the solution of the condenser principle are obtained solely in terms of equilibrium measures for suitable subsets. In particular, we get a formula for the effective resistance between any pair of vertices of a graph. Specifically, r xy = 1/n (v x (y) + v y (x)) , where v z denotes the equilibrium measure for the set V - {z} . In any case, the equilibrium measure for a proper subset is accomplished by solving a Linear Programming Problem.

The inverses of some circulant matrices

We present here necessary and sufficient conditions for the invertibility of some circulant matrices that depend on three parameters and moreover, we explicitly compute the inverse. Our study also encompasses a wide class of circulant symmetric matrices. The techniques we use are related with the solution of boundary value problems associated to second order linear difference equations. Consequently, we reduce the computational cost of the problem. In particular, we recover the inverses of some well known circulant matrices whose coefficients are arithmetic or geometric sequences, Horadam numbers among others. We also characterize when a general symmetric, circulant and tridiagonal matrix is invertible and in this case, we compute explicitly its inverse.

Distance–regular graphs having the M-property

We analyse when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is an M-matrix; that is, it has non-positive off-diagonal elements or, equivalently when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is also the combinatorial Laplacian of another network. When this occurs we say that the distance–regular graph has the M-property. We prove that only distance–regular graphs with diameter up to three can have the M-property and we give a characterization of the graphs that satisfy the M-property in terms of their intersection array. Moreover, we exhaustively analyse strongly regular graphs having the M-property and we give some families of distance–regular graphs with diameter three that satisfy the M-property. Roughly speaking, we prove that all distance–regular graphs with diameter one; about half of the strongly regular graphs; only some imprimitive distance–regular graphs with diameter three, and no distance–regular graphs with diameter greater than three, have the M-property. In addition, we conjecture that no primitive distance–regular graph with diameter three has the M-property.

Kirchhoff index of periodic linear chains

A periodic linear chain consists of a weighted

Boundary value problems on weighted networks

2n