In , a study of the existence and uniqueness of solution of partial overdetermined boundary value problems for finite networks was performed. These problems involve Schrodinger operators and the novel feature is that no data are prescribed on part of the boundary, whereas both the values of the function and of its normal derivative are given on another part of the boundary. In the present work, we study the resolvent kernels associated with overdetermined partial boundary value problems on finite network and we express them in terms of the well-known Green operator and the Dirichlet-to-Robin map. Moreover, we analyze their main properties and we compute them in the case of a generalized cylinder. The obtained expression involve polynomials that can be seen as a generalization of Chebyshev polynomials, and indeed when the conductances along axes are constant the expressions for the overdetermined partial resolvent kernels are given in terms of second kind Chebyshev polynomials. (C) 2015 Elsevier Inc. All rights reserved.
Any elliptic operator defines an automorphism on the orthogonal subspace to the eigenfunctions associated with the lowest eigenvalue, whose inverse is the orthogonal Green operator. In this study, we show that elliptic Schrödinger operators on networks that have been obtained by adding a new vertex to a given network, can be seen as perturbations of the Schrödinger operators on the initial network. Therefore, the Green function of the new network can be computed in terms of the Green function of the original network.
We present here necessary and sucient conditions for the invertibility of circulant and symmetric matrices that depend on three parameters and moreover, we explicitly compute the inverse. The techniques we use are related with the solution of boundary value problems associated to second order linear dierence equations. Consequently, we reduce the computational cost of the problem. In particular, we recover the inverses of some well known circulant matrices whose coecients 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.
We present here necessary and su cient conditions for the invertibility of circulant and symmetric matrices that depend on three parameters
and moreover, we explicitly compute the inverse. The techniques we use are related with the solution of boundary value problems associated to second order linear di erence equations. Consequently, we reduce the computational cost of the problem. In particular, we recover the inverses of some well known circulant matrices whose coe cients 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.
In this work, we consider circulant matrices of type A = Circ(a, -b, -c, . . . , -c, b). This type of matrices
raise, among others, when dealing for example, with the problem of computing the Green function of some
networks obtained by the addition of new vertices to a previously known one. We give a necessary and
sufficient condition for its invertibility. Moreover, as it is known, their inverse is a circulant matrix and we
explicitely give a closed formula for the expression of the coefficients.
Our aim is to characterize those matrices that are the response matrix of a semi positive definite Schrodinger operator on a circular planar network. Our findings generalize the known results and allow us to consider both nonsingular and Hon diagonally dominant matrices as response matrices. To this end, we define the Dirichlet-to-Robin map associated with a Schrodinger operator on general networks, and we prove that it satisfies the alternating property which is essential to characterize the response matrices.
We consider here the discrete analogue of Serrin's problem: if the equilibrium measure of a network with boundary satisfies that its normal derivative is constant, what can be said about the structure of the network and the symmetry of the equilibrium measure? In the original Serrin's problem, the conclusion is that the domain is a ball and the solution is radial. To study the discrete Serrin's problem, we first introduce the notion of radial function and then prove a generalization of the minimum principle, which is one of the main tools in the continuous case. Moreover, we obtain similar results to those of the continuous case for some families of networks with a ball-like structure, which include spider networks with radial conductances, distance-regular graphs or, more generally, regular layered networks.
A periodic linear chain consists of a weighted 2n -path where new edges have been added following a certain periodicity. In this paper, we obtain the effective resistance and the Kirchhoff index of a periodic linear chain as non trivial functions of the corresponding expressions for the path. We compute the expression of the Kirchhoff index of any homogeneous and periodic linear chain which generalizes the previously known results for ladder-like and hexagonal chains, that correspond to periods one and two respectively
A polyomino is an edge-connected union of cells in the planar
square lattice. Here we consider generalized linear polyominoes;
that is, the polyominoes supported by an n Ã— 2 lattice. In this
paper, we obtain the Green function and the Kirchhoff index of a
generalized linear polyomino as a perturbation of a 2n-path by adding
weighted edges between opposite vertices. This approach deeply links
generalized linear polyomino Green functions with the inverse M-matrix
problem, and especially with the so-called Green matrices.
Here, we consider a class of generalized linear chains; that is, the ladder-like chains as a perturbation of a 2n path by adding consecutive weighted edges between opposite vertices. This class of chains in particular includes a big family of networks that goes from the cycle, unicycle chains up to ladder networks. In this article, we obtain the Green function, the effective resistance, and the Kirchhoff index of ladder-like chains in terms of the Green function, the effective resistance, and the Kirchhoff index of the path. (c) 2014 Wiley Periodicals, Inc.
We survey here some techniques from Potential Theory and
show their use in electrical networks. We are focused on the linear
aspect of the theory; that is, on the analysis of singular an positive
semidefinite Schrödinger operators and we show the relation between
this class of operators and singular M-matrices, paying attention
on the properties of superharmonic functions. We also apply these
methods to establish some global results on the associated Markov
We consider a class of generalized linear chains, namely,
the unicycle chains as a perturbation of a 2nâ€“path by adding one
weighted edge between two specific vertices. This class of chains
includes, in particular, the weighted cycle. In this work, we obtain
the Green function, the effective resistance and the Kirchhoff index
of the unicycle chains as a function of the Green function, the effective
resistance and the Kirchhoff index of a path, respectively.
The Laplacian matrix of a simple graph has been widely studied, as a consequence of its applications. However the Laplacian matrix of a weighted graph is still a challenge. In this work we provide the Moore-Penrose inverse of the Laplacian matrix of the graph obtained adding new pendant vertices to an initial graph, in terms of the Moore-Penrose inverse of the Laplacian matrix of the original graph. As an application we can compute the effective resistances and the Kirchhoff index of the new network.
We consider a class of generalized linear chains, namely, the unicycle chains as a perturbation of a 2n–path by adding one weighted edge between two specific vertices. This class of chains includes, in particular, the weighted cycle. In this work, we obtain the Green function, the effective resistance and the Kirchhoff index of the unicycle chains as a function of the Green function, the effective resistance and the Kirchhoff index of a path, respectively
Aquesta tesi té dos objectius generals. Primer, volem deduir dades funcionals, estructurals o resistives d'una xarxa fent servir la informació proporcionada per la seva conductivitat. L'objectiu real és aconseguir aquesta informació d'una xarxa gran quan coneixem la mateixa de les subxarxes que la formen. El motiu és que les xarxes grans no són fàcils de treballar a causa de la seva mida. Com més petita sigui una xarxa, més fàcil serà treballar-hi, i per tant intentem trencar les xarxes grans en parts més petites que potser ens permeten resoldre problemes sobre elles més fàcilment. Principalment busquem les expressions de certs operadors que caracteritzen les solucions dels problemes de contorn en les xaxes originals. Aquests problemes es diuen problemes directes, ja que s'empren directament les dades de conductivitat per obtenir informació.El segon objectiu és recuperar les dades de conductivitat a l'interior d'una xarxa emprant només mesures a la frontera de la mateixa i condicions d'equiliri globals. Com que aquest problema no està ben establert perquè és altament sensible als canvis en les dades de frontera, de vegades només busquem una reconstrucció partial de la conductivitat o afegim condicions a la xarxa per tal de recuperar completament la conductivitat. Aquest tipus de problemes es diuen problemes inversos, ja que es fa servir informació a la frontera per aconseguir coneixements de l'interior de la xarxa. Aquest treball tracta de trobar situacions on la recuperació, total o parcial, es pugui dur a terme.Una de les nostres ambicions quant a problemes inversos és recuperar l'estructura de les xarxes per les que el ben conegut Problema de Serrin té solució en el camp discret. Sorprenentment, la resposta és similar al cas continu. També volem caracteritzar les xarxes mitjançant un operador a la frontera. Amb aquesta finalitat definim els problemes de contorn parcials sobredeterminats i describim com les solucions d'aquesta família de problemes que tenen una propietat d'alternància a una part de la frontera es propaguen a través de la xarxa mantenint aquesta alternància. Si ens centrem en una certa família de xarxes, veiem que l'operador a la frontera que abans hem mencionat pot ser la matriu de respostes d'una família infinita de xarxes amb diferentes conductivitats. Escollint una extensió en concret, obtenim una única xarxa per la qual una matriu donada és la seva matriu de respostes.Un cop hem caracteritzat aquelles matrius que són la matriu de respostes de certes xarxes, intentem recuperar les conductàncies d'aquestes xarxes. Amb aquesta finalitat, caracteritzem qualsevol solució d'un problema de contorn parcial sobredeterminat. Després, analitzem dos gran grups de xarxes que tene propietats de frontera notables i que ens porten a la recuperació de les conductàncies de certes branques a prop de la frontera. L'objectiu és donar fórmules explícites per obtenir aquestes conductàncies. Fent servir aquestes fórmules, aconseguim dur a terme una recuperació completa de conductàncies sota certes circumstàncies.
J. Serrin dio solución, en 1971, al siguiente problema de la Teoría del Potencial: sea O un dominio regular abierto, conexo y acotado del espacio Euclídeo tal que la solución u del problema de Dirichlet -¿u = 1 en O y u = 0 en d(O) viene sobredeterminada por la condición de que ¿u es constante en la frontera d(O). Entonces, O es una bola y u tiene simetría radial. Aquí consideramos el problema discreto análogo. Dado un grafo G = (V, E) y un subconjunto conexo F ¿ V, la solución del problema de Dirichlet
L(u) = 1 en F y u = 0 en d(F ) se llama medida de equilibrio. El Problema de Serrin Discreto se formula como: ¿la condición adicional ¿u = constante en d(F) implica que F es una bola y u es radial? La respuesta es negativa. Imponiendo una restricción que demostramos necesaria, los valores de la medida de equilibrio sí que dependen de la distancia a la frontera y esto nos lleva a dar una caracterización de los grafos donde el Problema de Serrin Discreto tiene solución.
Carmona, A.; Encinas, A.; Gago, S. Czech-Slovak International Symposium on Graph Theory, Combinatorics, Algorithms and Applications p. 27 Presentation's date: 2013-07-12 Presentation of work at congresses
Metrics in graphs provide measures of proximity between vertices. The classical short-path distances can be replaced for more general metrics, as the adjusted forest metric introduced by Chebotarev et al. in . Other related distance is the one provided for the resistance distance of a network . The objective of our work is to generalize the adjusted forest metric related to Laplacian operators to the adjusted forest metric related to Schr¨odinger operators, under the functional analysis framework. Furthermore, we show that it can be computed in terms ofthe effective resistances of the network.
In 1971, J. Serrin solved the following problem on Potential Theory: If is a smooth bounded open connected domain in the Euclidean space for which the solution u to the Dirichlet problem u = -1 in and u = 0 on ¿ has overdetermined boundary condition ¿u ¿n = constant on ¿ , then is a ball and u has radial symmetry. This result has multiple extensions, and what is more interesting, the methods used for solving the problem have applications to study the symmetry of the solutions of very general elliptic problems. We consider here the discrete analogue problem. Specifically, given ¿ = (V,E) a graph and F ¿ V a connected subset, the solution of the Dirichlet problem L(u) = 1 on F and u = 0 on d(F) is known as equilibrium measure. Therefore, the discrete Serrin’s problem can be formulated as: does the additional condition ¿u ¿n = constant on d(F) imply that F is a ball and u is radial? We provide some examples showing that the answer is negative. However, the values of the equilibrium measure depend on the distance to the boundary and hence we wonder if imposing some symmetries on the domain, the solution to Serrin’s problem is radial. The conclusion is true for many families of graph with high degree of symmetry, for instance for distance–regular graphs or spider networks.
A Polyomino is an edge–conected union of cells in the planar square lattice. Polyominoes are very popular in mathematical recreations, and have found interest among mathematicians, physicists, biologists, and computer scientists as well. Because the chemical constitution of a molecule is conventionally represented by a molecular graph or network, the polyominoes have deserved the attention of the Organic Chemistry community. So, several molecular structure descriptors based in network structural descriptors, have been introduced.
In particular, in the last decade a great amount of works devoted to calculate the Kirchhoff Index of linear polyominoes–like networks, have been published. In this work we deal with this class of polyominoes, that we call generalized linear polyominoes, that besides the
most popular class of linear polyomino chains, also includes cycles, Phenylenes and Hexagonal chains to name only a few. Because the Kirchhoff Index is the trace of the Green function of the network, here we obtain the Green function of generalized linear Polyominoes.
To do this, we understand a Polyomino as a perturbation of a path by adding weighted edges between opposite vertices. Therefore, unlike the techniques used by Yang and Zhang in 2008, that are based on the decomposition of the combinatorial Laplacian in structured blocks, here we obtain the Green function of a linear Polyomino from a perturbation of the combinatorial Laplacian. This approach deeply link linear Polyomino Green functions with the inverse M –matrix problem and specially, with the so–called Green matrices, of Gantmacher and Krein (1941).
In this work we concentrate on determining explicit expressions, via suitable orthogonal polynomials on the line, for the Green function associated with any regular boundary value problem on a weighted path, whose weights are determined by the coefficients of the three-term recurrence relation.
In this work, we concentrate on determining explicit expressions, via
suitable orthogonal polynomials on the line, for the Green function associated with
any regular boundary value problem on a weighted path, whose weights are determined
by the coefficients of the three-term recurrence relation.
Inverse boundary-value problems were born to answer the question of whether it is possible to determine the conductivity of a body by means of boundary measurements. These problems are exponentially ill-posed since its solutions are highly sensitive to changes in the boundary data. We are mainly interested on the discrete version of the problem, that is, the inverse boundary-value problems on finite weighted networks. The aim here is to study partial inverse boundary-value problems, which are characterized by the existence of a part of the boundary where no data is known.
Given a weighted network with conductances on the edges $\Gamma=(V,c)$, we fix a proper and connected subset $F\subset V$ and will consider a certain kind of boundary value problems in which the values of the functions and of their normal derivatives are known at the same part of the boundary of $F$ and there exists another part of the boundary where no data is known. We determine when there is existance and/or uniqueness of solution on $\bar F$. For, it is mandatory to consider the Dirichlet-to-Neumann map of the network, its kernel and a local inverse of the matrix given by this kernel. We also observe that the kernel of the Dirichlet-to-Neumann map is a Schur Complement of the Schrödinger operator of the network.
Regular boundary value problems on a distance-regular graph associated with Schrodinger operators are analyzed. These problems include the cases in which the boundary has one or two vertices. In each case, the Green matrices are given in terms of two families of orthogonal polynomials, one of them corresponding with the distance polynomials of the distance-regular graphs.
In this work we derive the Creen function of a generalized linear Polyomino as a suitable perturbation of the Creen function of a Hamiltonian path on it. So, our study encompasses previous work:; on polyomino-like chains.
In this work we analyze regular boundary value problems on a distanceregular
graph associated with Schr¨odinger operators in the case that the boundary has
two vertices. Moreover, we obtain the Green matrix for each regular problem. In each
case, the Green matrix is given in terms of two families of orthogonal polynomials, one
of them corresponding with the distance polynomials of the distance-regular graph.
Our objective is to determine the Green function of product networks in terms of the Green function of one of the factor networks and the eigenvalues and eigenfunctions of the Schr ¨odinger operator of the other factor network, which we consider that are known. Moreover, we use these results to obtain the Green function of spider networks in terms of Green functions over cicles and paths
Our objective is to determine the Green function of product networks in terms of the Green function of one of the factor networks and the eigenvalues and eigenfunctions of the Schr odinger operator of the other factor network, which we consider that are known. Moreover, we use these results to obtain the Green function of spider networks in terms of Green functions over cicles and paths.
The díscrete Green functions and their relationship whit discrete Laplace equations
have deserved the interest of many researchs useing different approac. In this work we derive the Green function of a perturbed network in terms of the Green function of its base network.